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. Who is expected to fail fewer times before singing a high C note for the first time?

Given that, The probabilities that Sylvie can sing a perfect high C note on each attempt is 4 7 The probabilities that Thierry can sing a perfect high C note on each attempt is 5 8 We know that, the expectation of the geometric distribution is reciprocal of p i.e., 1 p , where p is probability of success. E Sylvie = 1 4 7 = 7 4 = 1 . 75 , E Thierry = 1 5 8 = 8 5 = 1 . 6 Therefore, Thierry is expected to fail fewer times.

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. Find the probability that both Sylvie and Thierry succeed in singing a high C note on their second attempts.

Given that, The probabilities that Sylvie can sing a perfect high C note on each attempt is 4 7 The probabilities that Thierry can sing a perfect high C note on each attempt is 5 8 Let X , Y be the number of attempts made by Sylvie and Thierry To find P X = 2 × P Y = 2 : A random variable X that has a geometric distribution is denoted by X ~ Geo ( p ) , and the probability that the first success occurs on the r th trial is P X = r = p ( 1 − p ) r − 1 for r = 1 , 2 , 3 , … ⇒ P X = 2 × P Y = 2 = 4 7 × 3 7 2 - 1 × 5 8 × 3 8 2 - 1 ⇒ P X = 2 × P Y = 2 = 45 784 = 0 . 0574 .

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AS and A Level

IMPORTANT

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A biased $4$ -sided die is numbered $1, 3, 5$ and $7$ . The probability of obtaining each score is proportional to that score. Find the probability that the first prime number is obtained on the third roll of the die.

Important Questions on The Binomial and Geometric Distributions

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AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 7 - The Binomial and Geometric Distributions>EXERCISE 7D>Q 7

Sylvie and Thierry are members of a choir. The probabilities that they can sing a perfect high

C

note on each attempt are

\frac{4}{7}

and

\frac{5}{8}

, respectively. Who is expected to fail fewer times before singing a high

C

note for the first time?

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 7 - The Binomial and Geometric Distributions>EXERCISE 7D>Q 7

Sylvie and Thierry are members of a choir. The probabilities that they can sing a perfect high

C

note on each attempt are

\frac{4}{7}

and

\frac{5}{8}

, respectively. Find the probability that both Sylvie and Thierry succeed in singing a high

C

note on their second attempts.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 7 - The Binomial and Geometric Distributions>EXERCISE 7D>Q 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. Given that

X

follows a geometric distribution, describe the way in which the cards are selected, and give the reason for your answer.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 7 - The Binomial and Geometric Distributions>EXERCISE 7D>Q 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. Find the probability that:

X

is equal to

E (X)

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 7 - The Binomial and Geometric Distributions>EXERCISE 7D>Q 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. Find the probability that: neither of the first two cards selected is a heart and the first diamond is the third card selected.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 7 - The Binomial and Geometric Distributions>EXERCISE 7D>Q 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 (X \leq b) > 0.865

, find

E (X)

and find the smallest possible value of

b

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AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 7 - The Binomial and Geometric Distributions>EXERCISE 7D>Q 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 . 5^{1} + 0 . 5^{3} + 0 . 5^{5} + 0 . 5^{7} + \dots

Find the probability that Zane wins the game.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 7 - The Binomial and Geometric Distributions>EXERCISE 7D>Q 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 . 5^{1} + 0 . 5^{3} + 0 . 5^{5} + 0 . 5^{7} + \dots

Describe the sequence of results represented by the value

0 . 5^{5}

in this series. Find the probability that Anouar wins the game.

A biased 4 -sided die is numbered 1,3,5 and 7. The probability of obtaining each score is proportional to that score. Find the probability that the first prime number is obtained on the third roll of the die.

Important Questions on The Binomial and Geometric Distributions

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A biased $4$ -sided die is numbered $1, 3, 5$ and $7$ . The probability of obtaining each score is proportional to that score. Find the probability that the first prime number is obtained on the third roll of the die.