15 Two ordinary fair dice are rolled simultaneously. Find the probability of obtaining:
a the first double on the fourth roll
b the first pair of numbers with a sum of more than 10 before the 10th roll.
16 X ~ Geo(0.24) and Y ~ Geo(0.25) are two independent random variables. Find the probability that X Y+ =4.
PS
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Step 3: Subtract one equation from the other.
Step 4: If you have successfully managed steps 1, 2 and 3, you should need no help completing the proof!
One in four boxes of Zingo breakfast cereal contains a free toy. Let the random variable X be the number of boxes that a child opens, up to and including the one in which they find their first toy.
a Find the mode and the expectation of X.
b Interpret the two values found in part a in the context of this question.
Answer
a The mode of X is 1.
E( ) 1 1
4 4
1
= =
=
−
X p
b A child is most likely to find their first toy in the first box they open but, on average, a child will find their first toy in the fourth box that they open.
WORKED EXAMPLE 7.10
The variable is
X ~ Geo 1
4 . An answer written ‘in
context’ must refer to a specific situation; in this case, the situation described in the question.
TIP
The variable X follows a geometric distribution. Given that E(X)=321, find P(X >6).
Answer
= = =
X p p
E( ) 1 7
2, so 2 7 1 –2
7 5
= =7
q
P( 6)
5 7 0.133
6 6
> =
=
=
X q
WORKED EXAMPLE 7.11
We find the parameter p and then we find q.
We use P(X >r)=qr from Key point 7.4.
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Alternatively, we can use
<
= − =
= −
=
<
<
P(1 3)
P( 3) P( 1)
819 1331
3 11 456 1331
X
X X
TIP Given that X ~ Geo( ) and that p P(Xø3) 819
=1331, find:
a P(X >3) b P(1<Xø3). Answer
a P(X 3) 1 P(Xø3) 1 819
1331 512 1331
> = −
= −
=
b q Xø
q q
q p
1 – P( 3)
1 819 1331 1 819
1331 8
11and 3 11
3 3
3
=
= −
= −
= =
X X X
pq pq
P(1 3) P( 2) P( 3)
456
1331or 0.343
2
< ø = = + =
= +
= WORKED EXAMPLE 7.12
We use 1 –qr=P(X<r) to find q and p.
EXERCISE 7D
1 Given that X ~ Geo(0.36), find the exact value of E(X).
2 The random variable Y follows a geometric distribution. Given that P(Y= =1) 0.2, find E( ).Y 3 Given that S~ Geo( ) and that p E( )S 41
= 2, find P(S=2).
4 Let T be the number of times that a fair coin is tossed, up to and including the toss on which the first tail is obtained. Find the mode and the mean of T.
5 Let X be the number of times an ordinary fair die is rolled, up to and including the roll on which the first 6 is obtained. Find E(X) and evaluate P[X >E(X)].
6 A biased 4-sided die is numbered 1, 3, 5 and 7. The probability of obtaining each score is proportional to that score.
a Find the expected number of times that the die will be rolled, up to and including the roll on which the first non-prime number is obtained.
b Find the probability that the first prime number is obtained on the third roll of the die.
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7 Sylvie and Thierry are members of a choir. The probabilities that they can sing a perfect high C note on each attempt are 4
7 and 5
8, respectively.
a Who is expected to fail fewer times before singing a high C note for the first time?
b Find the probability that both Sylvie and Thierry succeed in singing a high C note on their second attempts.
8 A standard deck of 52 playing cards has an equal number of hearts, spades, clubs and diamonds. A deck is shuffled and a card is randomly selected. Let X be the number of cards selected, up to and including the first diamond.
a Given that X follows a geometric distribution, describe the way in which the cards are selected, and give the reason for your answer.
b Find the probability that:
i X is equal to E(X)
ii neither of the first two cards selected is a heart and the first diamond is the third card selected.
9 A study reports that a particular gene in 0.2% of all people is defective. X is the number of randomly selected people, up to and including the first person that has this defective gene. Given that P(Xøb)>0.865, find
X
E( ) and find the smallest possible value of b.
10 Anouar and Zane play a game in which they take turns at tossing a fair coin. The first person to toss heads is the winner. Anouar tosses the coin first, and the probability that he wins the game is
+ + + +…
0.51 0.53 0.55 0.57 .
a Describe the sequence of results represented by the value 0.55 in this series.
b Find, in a similar form, the probability that Zane wins the game.
c Find the probability that Anouar wins the game.
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In a game for two people that cannot be drawn, you are the stronger player with a 60% chance of winning each game.
The probability distributions for the number of games won by you and those won by your opponent when a single game is played, X and Y, are shown.
x 0 1 y 0 1
X==x
P( ) 0.40 0.60 P(Y= )y 0.60 0.40
Investigate the probability distributions for X and Y in a best-of-three contest, where the first player to win two games wins the contest.
Who gains the advantage as the number of games played in a contest increases? What evidence do you have to support your answer?
How likely are you to win a best-of-five contest?
PS
EXPLORE 7.4
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Checklist of learning and understanding
● A binomial distribution can be used to model the number of successes in a series of n repeated independent trials where the probability of success on each trial, p, is constant.
● If X~ B( , ) then n p =
p n
r p p
r r(1 – )n r–.
● E(X)= =µ np
● Var(X)=σ2=np(1 – )p =npq, whereq=1 – .p
● A geometric distribution can be used to model the number of trials up to and including the first success in a series of repeated independent trials where the probability of success on each trial, p, is constant.
● If X~ Geo( ) then p pr=p(1 – )pr–1 for r=1, 2, 3,…
● E(X) =µ = p 1
● P X( ør) 1 –= qr and P(X> =r) qr, where q=1 – .p
● The mode of all geometric distributions is 1.
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END-OF-CHAPTER REVIEW EXERCISE 7
1 Given that
X B n
~ ,n1
find an expression for P(X=1) in terms of n. [2]
2 A family has booked a long holiday in Skragness, where the probability of rain on any particular day is 0.3.
Find the probability that:
a the first day of rain is on the third day of their holiday [1]
b it does not rain for the first 2 weeks of their holiday. [2]
3 One plastic robot is given away free inside each packet of a certain brand of biscuits. There are four colours of plastic robot (red, yellow, blue and green) and each colour is equally likely to occur. Nick buys some packets of these biscuits. Find the probability that
i he gets a green robot on opening his first packet, [1]
ii he gets his first green robot on opening his fifth packet. [2]
Nick’s friend Amos is also collecting robots.
iii Find the probability that the first four packets Amos opens all contain different coloured robots. [3]
Cambridge International AS & A Level Mathematics 9709 Paper 62 Q3 November 2015 4 Weiqi has two fair triangular spinners. The sides of one spinner are labelled 1, 2, 3, and the sides of the other are
labelled 2, 3, 4. Weiqi spins them simultaneously and notes the two numbers on which they come to rest.
a Find the probability that these two numbers differ by 1. [2]
b Weiqi spins both spinners simultaneously on 15 occasions. Find the probability that the numbers on which they come to rest do not differ by 1 on exactly eight or nine of the 15 occasions. [3]
5 A computer generates random numbers using any of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The numbers appear on the screen in blocks of five digits, such as 50119 26317 40068 ... Find the probability that:
a there are no 7s in the first block [1]
b the first zero appears in the first block [1]
c the first 9 appears in the second block. [2]
6 Four ordinary fair dice are rolled.
a In how many ways can the four numbers obtained have a sum of 22? [2]
b Find the probability that the four numbers obtained have a sum of 22. [2]
c The four dice are rolled on eight occasions. Find the probability that the four numbers obtained have a
sum of 22 on at least two of these occasions. [3]
7 When a certain driver parks their car in the evenings, they are equally likely to remember or to forget to switch off the headlights. Giving your answers in their simplest index form, find the probability that on the next 16 occasions that they park their car in the evening, they forget to switch off the headlights:
a 14 more times than they remember to switch them off [2]
b at least 12 more times than they remember to switch them off. [3]
8 Gina has been observing students at a university. Her data indicate that 60% of the males and 70% of the females are wearing earphones at any given time. She decides to interview randomly selected students and to interview males and females alternately.
M
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a Use Gina’s observation data to find the probability that the first person not wearing earphones is the third male interviewed, given that she first interviews:
i a male [2]
ii a female [2]
iii a male who is wearing earphones. [2]
b State any assumptions made about the wearing of earphones in your calculations for part a. [1]
9 In Restaurant Bijoux 13% of customers rated the food as ‘poor’, 22% of customers rated the food as ‘satisfactory’ and 65% rated it as ‘good’. A random sample of 12 customers who went for a meal at Restaurant Bijoux was taken.
i Find the probability that more than 2 and fewer than 12 of them rated the food as ‘good’. [3]
On a separate occasion, a random sample of n customers who went for a meal at the restaurant was taken.
ii Find the smallest value of n for which the probability that at least 1 person will rate the food as ‘poor’ is
greater than 0.95. [3]
Cambridge International AS & A Level Mathematics 9709 Paper 62 Q3 June 2012 10 A biased coin is four times as likely to land heads up compared with tails up. The coin is tossed k times so that
the probability that it lands tails up on at least one occasion is greater than 99%. Find the least possible
value of k. [4]
11 Given that X~ B( , 0.4) and that n P(X= = ×1) k P(X=n– 1), express the constant k in terms of n, and
find the smallest value of n for which k>25. [5]
12 A book publisher has noted that, on average, one page in eight contains at least one spelling error, one page in five contains at least one punctuation error, and that these errors occur independently and at random. The publisher checks 480 randomly selected pages from various books for errors.
a How many pages are expected to contain at least one of both types of error? [2]
b Find the probability that:
i the first spelling error occurs after the 10th page [2]
ii the first punctuation error occurs before the 10th page [2]
iii the 10th page is the first to contain both types of error. [2]
13 Robert uses his calculator to generate 5 random integers between 1 and 9 inclusive.
i Find the probability that at least 2 of the 5 integers are less than or equal to 4. [3]
Robert now generates n random integers between 1 and 9 inclusive. The random variable X is the number of these n integers which are less than or equal to a certain integer k between 1 and 9 inclusive. It is given that the mean of X is 96 and the variance of X is 32.
ii Find the values of n and k. [4]
Cambridge International AS & A Level Mathematics 9709 Paper 62 Q4 June 2013 14 Anna, Bel and Chai take turns, in that order, at rolling an ordinary fair die. The first person to roll a 6 wins
the game.
Find the ratio P(Anna wins) : P(Bel wins) : P(Chai wins), giving your answer in its simplest form. [7]
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