Given that S ~ Geo p and that E S = 4 1 2 , find P S = 2 .

The expectation of the geometric distribution is reciprocal of p i.e., 1 p Given, E S = 4 1 2 = 9 2 ⇒ 1 p = 9 2 ⇒ p = 2 9 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 , … We have to find P S = 2 : ⇒ P S = 2 = 2 9 7 9 2 - 1 ⇒ P S = 2 = 14 81 .

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.

MEDIUM

AS and A Level

IMPORTANT

Earn 100

$X ~ Geo (0.24)$ and $Y ~ Geo (0.25)$ are two independent random variables. Find the probability that $X + Y = 4 .$

Important Questions on The Binomial and Geometric Distributions

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 1

Given that

X ~ Geo (0.36)

, find the exact value of

E (X)

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 2

The random variable

Y

follows a geometric distribution. Given that

P (Y = 1) = 0.2

, find

E (Y)

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 3

Given that

S ~ Geo (p)

and that

E (S) = 4 \frac{1}{2}

, find

P (S = 2) .

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 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

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 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)]

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 6

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 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.

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 6

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.

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

C

note for the first time?

X~Geo0.24 and Y~Geo0.25 are two independent random variables. Find the probability that X+Y=4.

Important Questions on The Binomial and Geometric Distributions

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$X ~ Geo (0.24)$ and $Y ~ Geo (0.25)$ are two independent random variables. Find the probability that $X + Y = 4 .$