IB Mathematics

`x`	`0`	`1`	`2`
`P (X = x)`	`1/10`	`a`	`b`

Question 1

Let `A` and `B` be independent events such that `P(A) = 0.3` and `P(B) = 0.8`.

(a) Find `P(A nn B)`.

(b) Find `P(A uu B)`.

(c) Are `A` and `B` mutually exclusive? Justify your answer.

(a) Independent `=> P(A nn B) = P(A) xx P(B) (= 0.3 xx 0.8)``= 0.24`

(b) `P(A uu B) = P(A) + P(B) - P(A nn B) (= 0.3 + 0.8 - 0.24)``= 0.86`

(c) No, with valid reason

e.g. `P(A nn B) != 0` or `P(A uu B) != P(A) + P(B)` or correct numerical equivalent

Question 2

The heights of a group of students are normally distributed with a mean of 160 cm and a standard deviation of 20 cm.

(a) A student is chosen at random. Find the probability that the student's height is greater than 180 cm.

(b) In this group of students, 11.9 % have heights less than d cm. Find the value of d .

(a) `z = (180 - 160) / 20 = 1`

`phi(1) = 0.8413`

`P("height" > 180) = 1 - 0.8413 = 0.159`

(b) `z = -1.1800`

Setting up equation ` -1.18 = (d - 160) / 20`

`d = 136`

Question 3

Srinivasa places the nine labelled balls shown below into a box.

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Srinivasa then chooses two balls at random, one at a time, from the box. The first ball is not replaced before he chooses the second.

Find the probability that the first ball chosen is
1. labelled A;
2. labelled A or labelled N. [2]
Find the probability that the second ball chosen is labelled A, given that the first ball chosen was labelled N. [2]
Find the probability that both balls chosen are labelled N. [2]

(a) (i) `3/9 (1/3, 0.333, 0.333333..., 33.3%)`

(ii) `5/9 (0.556, 0.555555..., 55.6%)`

(b) `3/8 (0.375, 37.5%)`

Note: Award (A1) for correct numerator, (A1) for correct denominator.

(c) `2/9 xx 1/8`

Note: Award (M1) for a correct compound probability calculation seen.

`2/72 (1/36, 0.0278, 0.027777..., 2.78%)`

Question 4

Part A

Three students, Kim, Ching Li and Jonathan each have a pack of cards, from which they select a card at random. Each card has a 0, 3, 4, or 9 printed on it.

(a) Kim states that the probability distribution for her pack of cards is as follows.

`x`	0	3	4	9
`P(X = x)`	0.3	0.45	0.2	0.35

Explain why Kim is incorrect.

(b) Ching Li correctly states that the probability distribution for her pack of cards is as follows.

`x`	0	3	4	9
`P(X = x)`	0.4	`k`	`2k`	0.3

Find the value of k.

(c) Jonathan correctly states that the probability distribution for his pack of cards is given by `P(X=x)= (x+1)/20`. One card is drawn at random from his pack.

(i) Calculate the probability that the number on the card drawn is 0.
(ii) Calculate the probability that the number on the card drawn is greater than 0.

Part B

A game is played, where a die is tossed and a marble selected from a bag.

Bag M contains 3 red marbles (R) and 2 green marbles (G).

Bag N contains 2 red marbles and 8 green marbles.

A fair six-sided die is tossed. If a 3 or 5 appears on the die, bag M is selected (M).

If any other number appears, bag N is selected (N).

A single marble is then drawn at random from the selected bag.

(a)

Copy and complete the probability tree diagram on your answer sheet.

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(b) (i) Write down the probability that bag M is selected and a green marble drawn from it.

(ii) Find the probability that a green marble is drawn from either bag.
(iii) Given that the marble is green, calculate the probability that it came from Bag M.

(c) A player wins $`2` for a red marble and $`5` for a green marble. What are his expected winnings?

Part A

(a) Adding probabilities

Evidence of knowing that sum `= 1` for probability distribution

e.g. Sum greater than `1`, sum `= 1.3`, sum does not equal `1`

(b) Equating sum to `1` (`3k + 0.7 = 1`

`k = 0.1`

(c) (i) `P(X = 0) = (0+1)/20 = 1/20`

(ii) Evidence of using

`P(X > 0) = 1 - P(X = 0)` (or `4/20 + 5/20 + 10/20`) `= 19/20`

PART B
(a)

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(b) (i) `P(M and G) = 1/3 xx 2/5 (= 2/15 = 0.133)`

(ii) `P(G) = 1/3 xx 2/5 + 2/3 xx 8/10 = 10/15 (= 2/3 = 0.667)`

(iii) `P(M | G) = (P(M cap G))/(P(G)) = (2/15)/(10/15) = 1/5` or `0.2`

(c) `P(R) = 1 - 2/3 = 1/3`

Evidence of using a correct formula

E (win) = `2 xx 1/3 + 5 xx 2/3`

(or `2 xx 1/3 xx 3/5 + 2 xx 2/3 xx 2/10 + 5 xx 1/3 xx 2/5 + 5 xx 2/3 xx 8/10`)

= $4 (accept `=12/3, 60/15)`

Question 5

In a class of 30 students, 19 play tennis, 3 play both tennis and volleyball, and 6 do not play either sport. The following Venn diagram shows the events “plays tennis” and “plays volleyball”. The values t and v represent numbers of students.

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(a) (i) Find the value of t .

(ii) Find the value of v .

(b) Find the probability that a randomly selected student from the class plays tennis or volleyball, but not both.

(a) (i) valid approach to find t

eg `t + 3 = 19, 19 - 3`

`t = 16` (may be seen on Venn diagram)

(ii) valid approach to find v

eg `t + 3 + v + 6 = 30, 30 - 19 - 6`

`v = 5` (may be seen on Venn diagram)

(b) valid approach

eg `16 + 5, 21` students, `1 - (3+6)/30`,

`21/30 (= 7/10)`

Question 6

In a class of 30 students, 18 are fluent in Spanish, 10 are fluent in French, and 5 are not fluent in either of these languages. The following Venn diagram shows the events “fluent in Spanish” and “fluent in French”. The values m, n, p and q represent numbers of students.

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(a) Write down the value of q .

(b) Find the value of n .

(c) Write down the value of m and of p .

(a) `q = 5`

(b) valid approach

eg ` (18 + 10 + 5) - 30, 28 - 25, 18 + 10 - n = 25`

`n = 3`

(c) valid approach for finding m or p (may be seen in part (b))

eg `18 - 3, 3 + p = 10`

`m = 15, p = 7`

Question 7

Claire rolls a six-sided die 16 times. The scores obtained are shown in the following frequency table.

Score	Frequency
1	p
2	q
3	4
4	2
5	0
6	3

It is given that the mean score is 3.

(a) Find the value of p and the value of q .

Each of Claire’s scores is multiplied by 10 in order to determine the final score for a game she is playing.

(b) Write down the mean final score.

(a) attempt to form equation for the sum of frequencies = 16 or mean = 3

`p + q + 4 + 2 + 3 = 16 (=> p + q = 7)`

` (p+2q+12+8+18) / 16 = 3 (=> p + 2q = 10)`

OR ` (p+2q+12+8+18) / (9+p+q) = 3 (=> 2p + q = 11)`

attempt to eliminate one variable from their equations

`p + 2(7 - p) + 38 = 48 OR 2(7 - q) + q = 11`

`p = 4` and `q = 3`

Note: Award `M1A0A0M0A1` for `p = 4, q = 3` with no working.

(b) mean final score = 30

Question 8

A bag contains buttons which are either red or blue.

Initially, the bag contains three red buttons and one blue button.

Francine randomly selects one button from the bag. She then replaces the button and adds one extra button of the same colour.

For example, if she selects a red button, she then replaces it and adds one extra red button so that the bag then contains four red buttons and one blue button.

Francine then randomly selects a second button from the bag.

The following tree diagram represents the probabilities of the first two selections.

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(a) Find the value of `p` and the value of `q` .

(b) Show that the probability that Francine selects two buttons of the same colour is `7/10` .Given that Francine selects two buttons of the same colour, find the probability that she selects two red buttons.

The random variable `X` is defined as the number of red buttons selected by Francine. The following table shows the probability distribution of `X`.

`x`	`0`	`1`	`2`
`P (X = x)`	`1/10`	`a`	`b`

(d) Find the value of `a` and the value of `b`.

(e) Hence, find the expected number of red buttons selected by FrancineFrancine restarts the process with three red buttons and one blue button in the bag. She selects buttons as before, replacing the button and adding one extra button of the same colour each time. She repeats this until she selects a blue button.

(f) Given that the first two buttons she selects are red, write down the probability that the next button she selects is blue.

The probability that she selects the first blue button after `n` selections in total is `3/56`.

(g) Find the value of `n` .

(a) evidence of understanding that there are now 3R and 2B

`p = 3/5, q = 2/5`

(b) attempt to add two products

`P(same) = P(R R or BB) = 3/4 xx 4/5 + 1/4 xx 2/5`

`= 14/20`

`= 7/10`

(c) attempt to use conditional probability formula in context

`P( R R | same) = (P( R R)) / (P(same))`

Note: Award `M0` if candidate only writes `P(A|B)` formula and nothing else.

` = (12/20) / (14/20) `

`= 12/14 ( = 6/7) `

(d) `a = 6/20 (= 3/10), b = 12/20 (= 6/10)`

(e)attempt to use the formula for `E(X)`

`E(X) = 0 xx 1/10 + 1 xx 6/20 + 2 xx 12/20`

`= 30/20 (= 3/2)`

(f) `1/6`

(g) `P(n-1 reds) = 3/4 xx 4/5 xx ... xx (n+1)/(n+2) (= 3/(n+2))`

`P(`next one blue`) = 1/(n+3)`

`P(n-` 1 reds then 1 blue`) = P(n-`1 reds`) xx P`(next one blue)

`3/(n+2) xx 1/(n+3) = 3/56`

`(n + 2)(n + 3) = 56`

`n = 5`

Note: If no working shown, award `M1A0A0A1` for `n = 5`.

Question 9

On a Monday at an amusement park, a sample of 40 visitors was randomly selected as they were leaving the park. They were asked how many times that day they had been on a ride called The Dragon. This information is summarized in the following frequency table.

Number of times on The Dragon	Frequency
0	6
1	16
2	13
3	2
4	3

It can be assumed that this sample is representative of all visitors to the park for the following day.

(a) For the following day, Tuesday, estimate

(i) the probability that a randomly selected visitor will ride The Dragon;

(ii) the expected number of times a visitor will ride The Dragon.

It is known that 1000 visitors will attend the amusement park on Tuesday. The Dragon can carry a maximum of 10 people each time it runs.

(c) Estimate the minimum number of times The Dragon must run to satisfy demand.

(a) (i) summing frequencies of riders or finding complement

probability=`34/40`

(ii) attempt to find expected value

`1640 + 2 xx 1340 + 3 xx 240 + 4 xx 340`

`16/40 + (2 xx 13/40) + (3 xx 2/40) + (4 xx 3/40)`

`60/40 (= 1.5)`

(b) evidence of their rides/visitor `xx 1000` or `-: 10`

`1500 OR 0.15`

`150` (times)

Question 10

Events `A` and `B` are such that `P(A) = 0.3` and `P(B) = 0.8`.

(a) Determine the value of `P(A nn B)` in the case where the events `A` and `B` are independent.

(b) Determine the value of `P(A uu B)` in the case where the events `A` and `B` are independent.

(c) Determine the maximum possible value of `P(A nn B)`, justifying your answer.

(a) `P(A nn B) = 0.24`

(b) `P(A uu B) = 1.1 - P(A nn B)`

`0 <= P(A uu B) <= 1`

Note: This may be conveyed in a clearly labelled diagram or written

explanation where `P(A uu B) = 1`

the minimum value of `P(A nn B)` is `0.1`

(c) `A` is a subset of `B` (so `P(A nn B) = P(A)`).

Question 1

Let `A` and `B` be independent events such that `P(A) = 0.3` and `P(B) = 0.8`.

(a) Find `P(A nn B)`.

(b) Find `P(A uu B)`.

(c) Are `A` and `B` mutually exclusive? Justify your answer.

Question 2

The heights of a group of students are normally distributed with a mean of 160 cm and a standard deviation of 20 cm.

(a) A student is chosen at random. Find the probability that the student's height is greater than 180 cm.

(b) In this group of students, 11.9 % have heights less than d cm. Find the value of d .

Question 3

Srinivasa places the nine labelled balls shown below into a box.

null

Srinivasa then chooses two balls at random, one at a time, from the box. The first ball is not replaced before he chooses the second.

Find the probability that the first ball chosen is
1. labelled A;
2. labelled A or labelled N. [2]
Find the probability that the second ball chosen is labelled A, given that the first ball chosen was labelled N. [2]
Find the probability that both balls chosen are labelled N. [2]

Question 4

Part A

Three students, Kim, Ching Li and Jonathan each have a pack of cards, from which they select a card at random. Each card has a 0, 3, 4, or 9 printed on it.

(a) Kim states that the probability distribution for her pack of cards is as follows.

`x`	0	3	4	9
`P(X = x)`	0.3	0.45	0.2	0.35

Explain why Kim is incorrect.

(b) Ching Li correctly states that the probability distribution for her pack of cards is as follows.

`x`	0	3	4	9
`P(X = x)`	0.4	`k`	`2k`	0.3

Find the value of k.

(c) Jonathan correctly states that the probability distribution for his pack of cards is given by `P(X=x)= (x+1)/20`. One card is drawn at random from his pack.

(i) Calculate the probability that the number on the card drawn is 0.
(ii) Calculate the probability that the number on the card drawn is greater than 0.

Part B

A game is played, where a die is tossed and a marble selected from a bag.

Bag M contains 3 red marbles (R) and 2 green marbles (G).

Bag N contains 2 red marbles and 8 green marbles.

A fair six-sided die is tossed. If a 3 or 5 appears on the die, bag M is selected (M).

If any other number appears, bag N is selected (N).

A single marble is then drawn at random from the selected bag.

(a)

Copy and complete the probability tree diagram on your answer sheet.

null

(b) (i) Write down the probability that bag M is selected and a green marble drawn from it.

(ii) Find the probability that a green marble is drawn from either bag.
(iii) Given that the marble is green, calculate the probability that it came from Bag M.

(c) A player wins $`2` for a red marble and $`5` for a green marble. What are his expected winnings?

Question 5

In a class of 30 students, 19 play tennis, 3 play both tennis and volleyball, and 6 do not play either sport. The following Venn diagram shows the events “plays tennis” and “plays volleyball”. The values t and v represent numbers of students.

null

(a) (i) Find the value of t .

(ii) Find the value of v .

(b) Find the probability that a randomly selected student from the class plays tennis or volleyball, but not both.

Question 6

In a class of 30 students, 18 are fluent in Spanish, 10 are fluent in French, and 5 are not fluent in either of these languages. The following Venn diagram shows the events “fluent in Spanish” and “fluent in French”. The values m, n, p and q represent numbers of students.

null

(a) Write down the value of q .

(b) Find the value of n .

(c) Write down the value of m and of p .

Question 7

Claire rolls a six-sided die 16 times. The scores obtained are shown in the following frequency table.

Score	Frequency
1	p
2	q
3	4
4	2
5	0
6	3

It is given that the mean score is 3.

(a) Find the value of p and the value of q .

Each of Claire’s scores is multiplied by 10 in order to determine the final score for a game she is playing.

(b) Write down the mean final score.

Question 8

A bag contains buttons which are either red or blue.

Initially, the bag contains three red buttons and one blue button.

Francine randomly selects one button from the bag. She then replaces the button and adds one extra button of the same colour.

For example, if she selects a red button, she then replaces it and adds one extra red button so that the bag then contains four red buttons and one blue button.

Francine then randomly selects a second button from the bag.

The following tree diagram represents the probabilities of the first two selections.

null

(a) Find the value of `p` and the value of `q` .

(b) Show that the probability that Francine selects two buttons of the same colour is `7/10` .Given that Francine selects two buttons of the same colour, find the probability that she selects two red buttons.

The random variable `X` is defined as the number of red buttons selected by Francine. The following table shows the probability distribution of `X`.

`x`	`0`	`1`	`2`
`P (X = x)`	`1/10`	`a`	`b`

(d) Find the value of `a` and the value of `b`.

(e) Hence, find the expected number of red buttons selected by FrancineFrancine restarts the process with three red buttons and one blue button in the bag. She selects buttons as before, replacing the button and adding one extra button of the same colour each time. She repeats this until she selects a blue button.

(f) Given that the first two buttons she selects are red, write down the probability that the next button she selects is blue.

The probability that she selects the first blue button after `n` selections in total is `3/56`.

(g) Find the value of `n` .

Question 9

On a Monday at an amusement park, a sample of 40 visitors was randomly selected as they were leaving the park. They were asked how many times that day they had been on a ride called The Dragon. This information is summarized in the following frequency table.

Number of times on The Dragon	Frequency
0	6
1	16
2	13
3	2
4	3

It can be assumed that this sample is representative of all visitors to the park for the following day.

(a) For the following day, Tuesday, estimate

(i) the probability that a randomly selected visitor will ride The Dragon;

(ii) the expected number of times a visitor will ride The Dragon.

It is known that 1000 visitors will attend the amusement park on Tuesday. The Dragon can carry a maximum of 10 people each time it runs.

(c) Estimate the minimum number of times The Dragon must run to satisfy demand.

Question 10

Events `A` and `B` are such that `P(A) = 0.3` and `P(B) = 0.8`.

(a) Determine the value of `P(A nn B)` in the case where the events `A` and `B` are independent.

(b) Determine the value of `P(A uu B)` in the case where the events `A` and `B` are independent.

(c) Determine the maximum possible value of `P(A nn B)`, justifying your answer.

4.3 Probability Basics

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