Question 1
A game is played where two unbiased dice are rolled and the score in the game is the greater of the two numbers shown. If the two numbers are the same, then the score in the game is the number shown on one of the dice. A diagram showing the possible outcomes is given below.
Let `T` be the random variable “the score in a game”.
(a) Complete the table to show the probability distribution of `T`.
(b) Find the probability that
(i) a player scores at least 3 in a game.
(ii) a player scores 6, given that they scored at least 3.
(c) Find the expected score of a game.
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Question 2
In a game, balls are thrown to hit a target. The random variable `X` is the number of times the target is hit in five attempts. The probability distribution for `X` is shown in the following table.
(a) Find the value of `k`
The player has a chance to win money based on how many times they hit the target.
The gain for the player, in $, is shown in the following table, where a negative gain means that the player loses money.
(b) Determine whether this game is fair. Justify your answer.
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Question 3
Gustav plays a game in which he first tosses an unbiased coin and then rolls an unbiased six-sided die.
If the coin shows tails, the score on the die is Gustav’s final number of points.
If the coin shows heads, one is added to the score on the die for Gustav’s final number of points.
(a) Find the probability that Gustav’s final number of points is 7.
(b) Complete the following table.
(c) Calculate the expected value of Gustav’s final number of points.
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Question 4
Zac raises funds for a library by running a game where players spin a needle. The final position of the needle results in an outcome where a player wins or loses money. The outcomes, with associated probabilities, are shown in the following diagram.
Let `X` represent the amount that a player of this game wins.
(a)
(i) Find the expected value of `X`.
(ii) Interpret your answer to part (a)(i).
To encourage a person to keep playing this game, Zac increases the winning prize for the second game they play from $5 to $6. For each successive game they play, the winning prize continues to increase by $1.
Emily plays `k` games. The `k^{th}` game is fair.
(b)
(i) Find the value of `k`.
(ii) Explain why Zac expects to raise money from the games Emily plays.
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Question 5
A biased four-sided die, A, is rolled. Let `X` be the score obtained when die A is rolled. The probability distribution for `X` is given in the following table.
(a) Find the value of `p`.
(b) Hence, find the value of `E(X)`.
A second biased four-sided die, B, is rolled. Let `Y` be the score obtained when die B is rolled. The probability distribution for `Y` is given in the following table.
(c)
(i) State the range of possible values of `r`.
(ii) Hence, find the range of possible values of `q`.
(d) Hence, find the range of possible values for `E(Y)`.
Agnes and Barbara play a game using these dice. Agnes rolls die A once and Barbara rolls die B once. The probability that Agnes’ score is less than Barbara’s score is `1/2`.
(e) Find the value of `E(Y)`.
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Question 6
A biased four-sided die with faces labelled 1, 2, 3 and 4 is rolled and the result recorded. Let `X` be the result obtained when the die is rolled. The probability distribution for `X` is given in the following table where `p` and `q` are constants.
For this probability distribution, it is known that `E(X)` = 2.
(a) Show that `p` = 0.4 and `q` = 0.2.
(b) Find `P(X > 2)`
Nicky plays a game with this four-sided die. In this game she is allowed a maximum of five rolls. Her score is calculated by adding the results of each roll. Nicky wins the game if her score is at least ten.
After three rolls of the die, Nicky has a score of four.
(c) Assuming that rolls of the die are independent, find the probability that Nicky wins the game.
David has two pairs of unbiased four-sided dice, a yellow pair and a red pair. Both yellow dice have faces labelled 1, 2, 3 and 4. Let `S` represent the sum obtained by
rolling the two yellow dice. The probability distribution for `S` is shown below.
The first red die has faces labelled 1, 2, 2 and 3. The second red die has faces labelled 1, `a`, `a` and `b`, where `a` < `b` and `a, b in ZZ^+`. The probability distribution for the sum obtained by rolling the red pair is the same as the distribution for the sum obtained by rolling the yellow pair.
(d) Determine the value of `b`.
(e) Find the value of `a`, providing evidence for your answer.
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Question 7
The following table shows the number of errors per page in a 100 page document.
(a) State whether the data is discrete, continuous or neither.
(b) Find the mean number of errors per page.
(c) Find the median number of errors per page.
(d) Write down the mode.
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Question 8
A random variable `X` has a probability distribution given in the following table.
(a) Determine the value of `E(X^2)`.
(b) Find the value of `Var(X)`.
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Question 9
A discrete random variable `X` has the following probability distribution.
(a) Find an expression for `q` in terms of `p`.
(b)
(i) Find the value of `p` which gives the largest value of `E(X)`.
(ii) Hence, find the largest value of `E(X)`.
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Question 10
The following table shows a probability distribution for the random variable `X`, where `E(X)=1.2`.
(a)
(i) Find `q`.
(ii) Find `p`.
A bag contains white and blue marbles, with at least three of each colour. Three marbles are drawn from the bag, without replacement. The number of blue marbles drawn is given by the random variable `X`.
(b)
(i) Write down the probability of drawing three blue marbles.
(ii) Explain why the probability of drawing three white marbles is `1/6`.
(iii) The bag contains a total of ten marbles of which `w` are white. Find `w`.
(c) Jill plays the game nine times. Find the probability that she wins exactly two prizes.
(d) Grant plays the game until he wins two prizes. Find the probability that he wins his second prize on his eighth attempt.
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Question 1
A game is played where two unbiased dice are rolled and the score in the game is the greater of the two numbers shown. If the two numbers are the same, then the score in the game is the number shown on one of the dice. A diagram showing the possible outcomes is given below.
Let `T` be the random variable “the score in a game”.
(a) Complete the table to show the probability distribution of `T`.
(b) Find the probability that
(i) a player scores at least 3 in a game.
(ii) a player scores 6, given that they scored at least 3.
(c) Find the expected score of a game.
(a)
(b)
(i) `32/36 \ (8/9, 0.888888..., 88.9%)`
(ii) use of conditional probability
e.g. denominator of 32 OR denominator of 0.888888…, etc.
`11/32` (0.34375, 34.4%)
(c)
`frac{1 times 1 +3 times 2 + 5 times 3+...+ 11 times 6}{36}=161/36 \ (4 \frac{17}{36}, 4.47, 4.47222...)`
Question 2
In a game, balls are thrown to hit a target. The random variable `X` is the number of times the target is hit in five attempts. The probability distribution for `X` is shown in the following table.
(a) Find the value of `k`
The player has a chance to win money based on how many times they hit the target.
The gain for the player, in $, is shown in the following table, where a negative gain means that the player loses money.
(b) Determine whether this game is fair. Justify your answer.
(a) 0.15 + 0.2 + `k` + 0.16 + 2`k` + 0.25 = 1
`k` = 0.08
(b) `(-4 times 0.15) + ( -3 times 0.2) + ( -1 times 0.08) + (0 times 0.16) + (1 times 0.16) + (4 times 0.25) = -0.12`
`E(X) ne 0` therefore the game is not fair
Question 3
Gustav plays a game in which he first tosses an unbiased coin and then rolls an unbiased six-sided die.
If the coin shows tails, the score on the die is Gustav’s final number of points.
If the coin shows heads, one is added to the score on the die for Gustav’s final number of points.
(a) Find the probability that Gustav’s final number of points is 7.
(b) Complete the following table.
(c) Calculate the expected value of Gustav’s final number of points.
(a) recognizing that only way to score 7 is to achieve a head and a 6 on die
e.g. `1/6` and `1/2` seen in an attempt to combine probabilities
`(1/6 times 1/2=)1/12 \ (0.0833333...)`
(b) there are two ways to score (e.g.) 5
achieve a head and a 4 on die, or a tail and a 5 on die
`(2(1/6 times 1/2)=)2/12 \ (1/6, 0.167, 0.16666...)`
(c) recognizing the probabilities in the table are symmetric
then expected value = 4
Question 4
Zac raises funds for a library by running a game where players spin a needle. The final position of the needle results in an outcome where a player wins or loses money. The outcomes, with associated probabilities, are shown in the following diagram.
Let `X` represent the amount that a player of this game wins.
(a)
(i) Find the expected value of `X`.
(ii) Interpret your answer to part (a)(i).
To encourage a person to keep playing this game, Zac increases the winning prize for the second game they play from $5 to $6. For each successive game they play, the winning prize continues to increase by $1.
Emily plays `k` games. The `k^{th}` game is fair.
(b)
(i) Find the value of `k`.
(ii) Explain why Zac expects to raise money from the games Emily plays.
(a)
(i) use of expected value formula.
`E(X) = 5 times 0.40 + (-8) times 0.1 + (-5) times 0.2 + (-10) times 0.3`
($) `-`2.8
(ii) Anyone of the following:
Do not accept:
(b)
(i) `E(X) = 0`
(calculation of winnings to make the game fair)
`(w times 0.40 + (-8) times 0.1 + (-5) times 0.2 + (-10) times 0.3=0)`
(`w` =) ($) 12
evidence of increase in winnings per game up to $12
$5, $6, $7, ... $12
`k` = 8 (games)
(ii) `E(X) < 0` for each (any) of the first 7 games (or equivalent)
Question 5
A biased four-sided die, A, is rolled. Let `X` be the score obtained when die A is rolled. The probability distribution for `X` is given in the following table.
(a) Find the value of `p`.
(b) Hence, find the value of `E(X)`.
A second biased four-sided die, B, is rolled. Let `Y` be the score obtained when die B is rolled. The probability distribution for `Y` is given in the following table.
(c)
(i) State the range of possible values of `r`.
(ii) Hence, find the range of possible values of `q`.
(d) Hence, find the range of possible values for `E(Y)`.
Agnes and Barbara play a game using these dice. Agnes rolls die A once and Barbara rolls die B once. The probability that Agnes’ score is less than Barbara’s score is `1/2`.
(e) Find the value of `E(Y)`.
(a) recognizing probabilities sum to 1
`p+p+p+1/2 p=1`
`p=2/7`
(b) valid attempt to find `E(X)`.
`1 times p + 2 times p + 3 times p + 4 times 1/2 p \ (=8p)`
`E(X) = 16/7`
(c)
(i) `0 <= r <= 1`
(ii) attempt to find a value of `q`
`0 <= 1 -3q <= 1`
`0 <= q <= 1/3`
(d) `E(Y) = 1 times q + 2 times q + 3 times q + 4 times r \ (= 2 + 2r or 4-6q)`
one correct boundary value
2 + 2(0) (= 2) or 2 + 2(1) (= 4)
`2 <= E(Y) <= 4`
(e) evidence of choosing at least four correct outcomes from 1&2, 1&3, 1&4, 2&3, 2&4, 3&4
`6/7 q + 6/7 r`
rearrange to make `q` the subject
`q = frac{4-E(Y)}{6}`
`3pq + 3p(1-3q)=1/2`
`6/7 \times (frac{4-E(Y)}{6}) + 6/7 times (1-3 (frac{4-E(Y)}{6}))=1/2`
`frac{2(E(Y)-1)}{7}=1/2`
`E(Y)=11/4`
Question 6
A biased four-sided die with faces labelled 1, 2, 3 and 4 is rolled and the result recorded. Let `X` be the result obtained when the die is rolled. The probability distribution for `X` is given in the following table where `p` and `q` are constants.
For this probability distribution, it is known that `E(X)` = 2.
(a) Show that `p` = 0.4 and `q` = 0.2.
(b) Find `P(X > 2)`
Nicky plays a game with this four-sided die. In this game she is allowed a maximum of five rolls. Her score is calculated by adding the results of each roll. Nicky wins the game if her score is at least ten.
After three rolls of the die, Nicky has a score of four.
(c) Assuming that rolls of the die are independent, find the probability that Nicky wins the game.
David has two pairs of unbiased four-sided dice, a yellow pair and a red pair. Both yellow dice have faces labelled 1, 2, 3 and 4. Let `S` represent the sum obtained by
rolling the two yellow dice. The probability distribution for `S` is shown below.
The first red die has faces labelled 1, 2, 2 and 3. The second red die has faces labelled 1, `a`, `a` and `b`, where `a` < `b` and `a, b in ZZ^+`. The probability distribution for the sum obtained by rolling the red pair is the same as the distribution for the sum obtained by rolling the yellow pair.
(d) Determine the value of `b`.
(e) Find the value of `a`, providing evidence for your answer.
(a) uses `sum P(X=x) = 1` to form a linear equation in `p` and `q`
correct equation in terms of `p` and `q` from summing to 1
`p` + `q` = 0.6
uses `E(X)` = 2 to form a linear equation in `p` and `q`
correct equation in terms of `p` and `q` from `E(X)` = 2
`p` + 3`q` = 1
evidence of correctly solving these equations simultaneously
for example, 2`q` = 0.4 `=> q` = 0.2
so `p` = 0.4 and `q` = 0.2
(b) valid approach
`P(X > 2) = P(X=3) + P(X=4) = 0.3`
(c) recognises at least one of the valid scores (6, 7, or 8) required to win the game
let `T` represent the score on the last two rolls
a score of 6 is obtained by rolling (2,4), (4,2) or (3,3)
`P(T=6) =` 2(0.3)(0.1) + (0.2)2 (`=` 0.1)
a score of 7 is obtained by rolling (3,4) or (4,3)
`P(T=7) =` 2(0.2)(0.1) (`=` 0.04)
a score of 8 is obtained by rolling (4,4)
`P(T=8) =` (0.1)2 (`=` 0.01)
`P(`Nicky wins`)=` 0.1 + 0.04 + 0.01 `=` 0.15
(d) 3 + `b` = 8
`b` = 5
(e) `P(S=5)=4/16`
`P(S=a+2)=4/16`
`=> a+ 2=5`
Then `a=3`
Question 7
The following table shows the number of errors per page in a 100 page document.
(a) State whether the data is discrete, continuous or neither.
(b) Find the mean number of errors per page.
(c) Find the median number of errors per page.
(d) Write down the mode.
(a) Discrete
(b) `frac{0+24+40+51+44}{100}=159/100=1.59`
(c) 1
(d) 0
Question 8
A random variable `X` has a probability distribution given in the following table.
(a) Determine the value of `E(X^2)`.
(b) Find the value of `Var(X)`.
(a) `E(X^2) = sum x^2 cdot P(X=x) = 10.37 (=10.4 \ \ \ 3 \s f)`
(b) `sd(X)=1.44069...`
`Var(X)=2.08 \ (=2.0756)`
Question 9
A discrete random variable `X` has the following probability distribution.
(a) Find an expression for `q` in terms of `p`.
(b)
(i) Find the value of `p` which gives the largest value of `E(X)`.
(ii) Hence, find the largest value of `E(X)`.
(a) evidence of summing probabilities to 1
e.g. `q+4p^2+p+0.7-4p^2=1, \ 1-4p^2-p-0.7+4p^2`
`q=0.3-p`
(b)
(i)
correct substitution into `E(X)` formula
e.g. `0 times (0.3-p)+1 times 4p^2+ 2 times p + 3 times (0.7-4p^2)`
valid approach to find when `E(X)` is a maximum
max on sketch of `E(X)`, `8p+2+3 times (-8p)=0`, `-b/{2a} = {-2}/{2 times (-8)}`
`p=1/8 \ \ (=0.125)` (exact) (accept `x=1/8`)
(ii) 2.225
`89/40` (exact), 2.23
Question 10
The following table shows a probability distribution for the random variable `X`, where `E(X)=1.2`.
(a)
(i) Find `q`.
(ii) Find `p`.
A bag contains white and blue marbles, with at least three of each colour. Three marbles are drawn from the bag, without replacement. The number of blue marbles drawn is given by the random variable `X`.
(b)
(i) Write down the probability of drawing three blue marbles.
(ii) Explain why the probability of drawing three white marbles is `1/6`.
(iii) The bag contains a total of ten marbles of which `w` are white. Find `w`.
(c) Jill plays the game nine times. Find the probability that she wins exactly two prizes.
(d) Grant plays the game until he wins two prizes. Find the probability that he wins his second prize on his eighth attempt.
(a)
(i) correct substitution into `E(X)` formula
e.g. `0(p)+1(0.5)+2(0.3)+3(q)=1.2`
`q=1/{30}, 0.0333`
(ii) evidence of summing probabilities to 1
e.g. `p+0.5+0.3+q=1`
`p=1/6, 0.167`
(b)
(i) `P`(3 blue) = `1/30, 0.0333`
(ii) valid reasoning
e.g. `P`(3 white) = `P`(0 blue) = `1/6`
(iii) valid method
e.g. `P`(3 white) = `w/10 * (w-1)/9 * (w-2)/8`
correct equation
e.g. `w/10 * (w-1)/9 * (w-2)/8=1/6`
`w=6`
(c) valid approach
e.g. `B(n, p), ((n),(r)) p^r q^(n-r), (0.167)^2 (0.833)^7, ((9), (2))`
0.279081
0.279
(d) recognizing one prize in first seven attempts
e.g. `((7),(1)), (1/6)^1 (5/6)^6`
correct working
e.g. `((7),(1))(1/6)^1 (5/6)^6, 0.390714`
correct approach
e.g. `((7),(1))(1/6)^1 (5/6)^6 times 1/6`
0.065119
0.0651
Question 1
A game is played where two unbiased dice are rolled and the score in the game is the greater of the two numbers shown. If the two numbers are the same, then the score in the game is the number shown on one of the dice. A diagram showing the possible outcomes is given below.
Let `T` be the random variable “the score in a game”.
(a) Complete the table to show the probability distribution of `T`.
(b) Find the probability that
(i) a player scores at least 3 in a game.
(ii) a player scores 6, given that they scored at least 3.
(c) Find the expected score of a game.
Question 2
In a game, balls are thrown to hit a target. The random variable `X` is the number of times the target is hit in five attempts. The probability distribution for `X` is shown in the following table.
(a) Find the value of `k`
The player has a chance to win money based on how many times they hit the target.
The gain for the player, in $, is shown in the following table, where a negative gain means that the player loses money.
(b) Determine whether this game is fair. Justify your answer.
Question 3
Gustav plays a game in which he first tosses an unbiased coin and then rolls an unbiased six-sided die.
If the coin shows tails, the score on the die is Gustav’s final number of points.
If the coin shows heads, one is added to the score on the die for Gustav’s final number of points.
(a) Find the probability that Gustav’s final number of points is 7.
(b) Complete the following table.
(c) Calculate the expected value of Gustav’s final number of points.
Question 4
Zac raises funds for a library by running a game where players spin a needle. The final position of the needle results in an outcome where a player wins or loses money. The outcomes, with associated probabilities, are shown in the following diagram.
Let `X` represent the amount that a player of this game wins.
(a)
(i) Find the expected value of `X`.
(ii) Interpret your answer to part (a)(i).
To encourage a person to keep playing this game, Zac increases the winning prize for the second game they play from $5 to $6. For each successive game they play, the winning prize continues to increase by $1.
Emily plays `k` games. The `k^{th}` game is fair.
(b)
(i) Find the value of `k`.
(ii) Explain why Zac expects to raise money from the games Emily plays.
Question 5
A biased four-sided die, A, is rolled. Let `X` be the score obtained when die A is rolled. The probability distribution for `X` is given in the following table.
(a) Find the value of `p`.
(b) Hence, find the value of `E(X)`.
A second biased four-sided die, B, is rolled. Let `Y` be the score obtained when die B is rolled. The probability distribution for `Y` is given in the following table.
(c)
(i) State the range of possible values of `r`.
(ii) Hence, find the range of possible values of `q`.
(d) Hence, find the range of possible values for `E(Y)`.
Agnes and Barbara play a game using these dice. Agnes rolls die A once and Barbara rolls die B once. The probability that Agnes’ score is less than Barbara’s score is `1/2`.
(e) Find the value of `E(Y)`.
Question 6
A biased four-sided die with faces labelled 1, 2, 3 and 4 is rolled and the result recorded. Let `X` be the result obtained when the die is rolled. The probability distribution for `X` is given in the following table where `p` and `q` are constants.
For this probability distribution, it is known that `E(X)` = 2.
(a) Show that `p` = 0.4 and `q` = 0.2.
(b) Find `P(X > 2)`
Nicky plays a game with this four-sided die. In this game she is allowed a maximum of five rolls. Her score is calculated by adding the results of each roll. Nicky wins the game if her score is at least ten.
After three rolls of the die, Nicky has a score of four.
(c) Assuming that rolls of the die are independent, find the probability that Nicky wins the game.
David has two pairs of unbiased four-sided dice, a yellow pair and a red pair. Both yellow dice have faces labelled 1, 2, 3 and 4. Let `S` represent the sum obtained by
rolling the two yellow dice. The probability distribution for `S` is shown below.
The first red die has faces labelled 1, 2, 2 and 3. The second red die has faces labelled 1, `a`, `a` and `b`, where `a` < `b` and `a, b in ZZ^+`. The probability distribution for the sum obtained by rolling the red pair is the same as the distribution for the sum obtained by rolling the yellow pair.
(d) Determine the value of `b`.
(e) Find the value of `a`, providing evidence for your answer.
Question 7
The following table shows the number of errors per page in a 100 page document.
(a) State whether the data is discrete, continuous or neither.
(b) Find the mean number of errors per page.
(c) Find the median number of errors per page.
(d) Write down the mode.
Question 8
A random variable `X` has a probability distribution given in the following table.
(a) Determine the value of `E(X^2)`.
(b) Find the value of `Var(X)`.
Question 9
A discrete random variable `X` has the following probability distribution.
(a) Find an expression for `q` in terms of `p`.
(b)
(i) Find the value of `p` which gives the largest value of `E(X)`.
(ii) Hence, find the largest value of `E(X)`.
Question 10
The following table shows a probability distribution for the random variable `X`, where `E(X)=1.2`.
(a)
(i) Find `q`.
(ii) Find `p`.
A bag contains white and blue marbles, with at least three of each colour. Three marbles are drawn from the bag, without replacement. The number of blue marbles drawn is given by the random variable `X`.
(b)
(i) Write down the probability of drawing three blue marbles.
(ii) Explain why the probability of drawing three white marbles is `1/6`.
(iii) The bag contains a total of ten marbles of which `w` are white. Find `w`.
(c) Jill plays the game nine times. Find the probability that she wins exactly two prizes.
(d) Grant plays the game until he wins two prizes. Find the probability that he wins his second prize on his eighth attempt.
