A biased coin is tossed three times. The probability distribution for H , the number of heads obtained, is shown in the following table. h 0 1 2 3 P ( H = h ) 0 . 512 0 . 384 0 . 096 a Give another discrete random variable that is related to these trials, and calculate the probability that its value is greater than the value of H .

Given that, Biased coin tossed three times :- Let, H be the Number of head obtained, h 0 1 2 3 P H = h 0 . 512 0 . 384 0 . 096 a We know, P ( h = 0 ) + P ( h = 1 ) + P ( h = 2 ) + P ( h = 3 ) = 1 ∴ a = 0 . 008 ∴ p ( heads ) = p ( h = 0 ) + p ( h = 2 ) + p ( h = 3 ) - P ( h = 1 ) = 0 . 512 + 0 . 096 + 0 . 008 - 0 . 384 = 0 . 232 ∴ p ( heads ) = 0 . 2 Let, T be the Number of tail obtained ∴ P ( T > H ) = P ( h = 0 ) + P ( h = 1 ) = 0 . 512 + 0 . 384 ∴ P ( T > H ) = 0 . 896

The discrete random variable R is such that R ∈ 1 , 3 , 5 , 7 Given that P R = r = k r + 1 r + 2 , find the value of the constant k .

Given that, R ∈ { 1 , 3 , 5 , 7 } ⇒ R = 1 , P ( R = 1 ) = k × 2 3 = 2 k 3 ⇒ R = 3 , P ( R = 3 ) = k × 4 5 = 4 k 5 ⇒ R = 5 , P ( R = 5 ) = k × 6 7 = 6 k 7 ⇒ R = 7 , P ( R = 7 ) = k × 8 9 = 8 k 9 Now, P ( R = 1 ) + P ( R = 3 ) + P ( R = 5 ) + P ( R = 7 ) = 1 ⇒ 2 k 3 + 4 k 5 + 6 k 7 + 8 k 9 = 1 ⇒ k = 315 1012 The value of constant k is 315 1012 .

The discrete random variable R is such that R ∈ 1 , 3 , 5 , 7 Given that P R = r = k r + 1 r + 2 , find the value of the constant k . Hence, find P R ⩽ 4 .

Given that, R ∈ { 1 , 3 , 5 , 7 } ⇒ R = 1 , P ( r = 1 ) = k × 2 3 = 2 k 3 ⇒ R = 3 , P ( r = 3 ) = k × 4 5 = 4 k 5 ⇒ R = 5 , P ( r = 5 ) = k × 6 7 = 6 k 7 ⇒ R = 7 , P ( r = 7 ) = k × 8 9 = 8 k 9 Now, P ( r = 1 ) + P ( r = 3 ) + P ( r = 5 ) + P ( r = 7 ) = 1 ⇒ 2 k 3 + 4 k 5 + 6 k 7 + 8 k 9 = 1 ∴ 210 k + 252 k + 270 k + 280 k 315 = 1 ⇒ k = 315 1012 ⇒ P ( R ⩽ 4 ) = P ( r = 1 ) + P ( r = 3 ) = 2 k 3 + 4 k 5 = 22 k 15 = 22 15 × 315 1012 ∴ P ( R ⩽ 4 ) = 21 46

The probability distribution for the random variable X is given in the following table. y 0 1 2 3 P X = x 0 . 10 0 . 12 0 . 36 0 . 42 Calculate E X and Var X

Given that, The Probability distribution for the random variable X is given in the following table, x 0 1 2 3 P X = x 0 . 1 0 . 12 0 . 36 0 . 42 ∴ E ( x ) = ∑ i = 1 n x i × P i = x 1 × P 1 + x 2 × P 2 + x 3 × P 3 + x 4 × P 4 = 0 × 0 . 1 + 1 × 0 . 12 + 2 × 0 . 36 + 3 × 0 . 42 = 0 + 0 . 12 + 0 . 72 + 1 . 26 ∴ E ( x ) = 2 · 1 Now, Var ( x ) = E x 2 - [ E ( x ) ] 2 Va r ( x ) = ∑ i = 1 n x i P i 2 - ∑ i = 1 n x i P i 2 ∑ l = 1 n x i 2 P i = ( 0 ) 2 × 0 . 1 + ( 1 ) 2 × 0 . 12 + ( 2 ) 2 × 0 . 36 + ( 3 ) 2 × 0 . 42 = 0 + 0 . 12 + 4 × 0 . 36 + 9 × 0 . 42 ⇒ ∑ l = 1 n x i 2 P i = 5 . 34 from (1) V ar ( x ) = 5 . 34 - ( 2 · 1 ) 2 = 5 . 34 - 4 . 41 ⇒ V ar ( x ) = 0 . 93

The probability distribution for the random variable Y is given in the following table. y 0 1 2 3 4 P Y = y 0 . 03 2 p 0 . 32 p 0 . 05 Find the value of p .

Given that, The probability distribution for the random variable Y is given in the following table. y 0 1 2 3 4 P Y = y 0 . 03 2 p 0 . 32 p 0 . 05 We know that, ⇒ P ( y = 0 ) + P ( Y = 1 ) + p ( 4 = 2 ) + P ( y = 3 ) + P ( y = 4 ) = 1 ⇒ 0 . 03 + 2 p + 0 . 32 + P + 0 . 05 = 1 ⇒ 3 p = 0 . 6 ∴ p = 0 . 2

The probability distribution for the random variable Y is given in the following table. y 0 1 2 3 4 P Y = y 0 . 03 2 p 0 . 32 p 0 . 05 Calculate E Y and the standard deviation of Y .

Given that, The probability distribution for the random variable Y is given in the following table. y 0 1 2 3 4 P Y = y 0 . 03 2 p 0 . 32 p 0 . 05 a) We know that, ⇒ P ( y = 0 ) + P ( Y = 1 ) + p ( 4 = 2 ) + P ( y = 3 ) + P ( y = 4 ) = 1 ⇒ 0 . 03 + 2 p + 0 . 32 + P + 0 . 05 = 1 ⇒ 3 p = 0 . 6 ∴ p = 0 . 2 y p y 2 p y p y 2 0 0 . 03 0 0 0 1 2 p 1 2 p 2 p 2 0 . 32 4 0 . 64 1 . 28 3 p 9 3 p 9 p 4 0 . 05 16 0 . 2 0 . 8 ∑ p y = 1 . 84 ∑ p y 2 = 4 . 28 E ( x ) = ∑ i = 1 n p i y i = P 1 × y 1 + P 2 × y 2 + P 3 × y 3 + P 4 × y 4 + P 5 × y 5 ⇒ E ( x ) = 1 . 84 Var ( x ) = ∑ i = 1 n p i y i 2 - ∑ E = 1 n p i y i 2 = 4 . 28 - ( 1 . 84 ) 2 ⇒ Var ( x ) = 0 . 895 Standard deviation = V a r ( x ) = 0 . 895 ∴ Standard deviation = 0 . 946

EASY

AS and A Level

IMPORTANT

Earn 100

A biased coin is tossed three times. The probability distribution for $H,$ the number of heads obtained, is shown in the following table.

$h$ $0$ $1$ $2$ $3$

$P (H = h)$ $0.512$ $0.384$ $0.096$ $a$

Find the probability of obtaining a head each time the coin is tossed.

Important Questions on Probability Distributions

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 6 - Probability Distributions>EXERCISE 6A>Q 16

A biased coin is tossed three times. The probability distribution for $H,$ the number of heads obtained, is shown in the following table.

$h$	$0$	$1$	$2$	$3$
$P (H = h)$	$0.512$	$0.384$	$0.096$	$a$

Give another discrete random variable that is related to these trials, and calculate the probability that its value is greater than the value of $H .$

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 6 - Probability Distributions>EXERCISE 6A>Q 17

Two ordinary fair dice are rolled. A score of $3$ points is awarded if exactly one die shows an odd number and there is also a difference of $1$ between the two numbers obtained. A player who rolls two even numbers is awarded a score of $2$ points, otherwise a player scores $1$ point.

Draw up the probability distribution table for $S,$ the number of points awarded.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 6 - Probability Distributions>EXERCISE 6A>Q 17

Find the probability that a player scores $3$ points, given that the sum of the numbers on their two dice is greater than $9 .$

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 6 - Probability Distributions>EXERCISE 6A>Q 18

The discrete random variable $R$ is such that $R \in \{1, 3, 5, 7\}$

Given that $P (R = r) = \frac{k (r + 1)}{r + 2},$ find the value of the constant $k .$

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 6 - Probability Distributions>EXERCISE 6A>Q 18

The discrete random variable $R$ is such that $R \in \{1, 3, 5, 7\}$

Given that $P (R = r) = \frac{k (r + 1)}{r + 2},$ find the value of the constant $k .$ Hence, find $P (R ⩽ 4) .$

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 6 - Probability Distributions>EXERCISE 6B>Q 1

The probability distribution for the random variable $X$ is given in the following table.

$y$	$0$	$1$	$2$	$3$
$P (X = x)$	$0.10$	$0.12$	$0.36$	$0.42$

Calculate $E (X)$ and $Var (X)$

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 6 - Probability Distributions>EXERCISE 6B>Q 2

The probability distribution for the random variable $Y$ is given in the following table.

$y$	$0$	$1$	$2$	$3$	$4$
$P (Y = y)$	$0.03$	$2 p$	$0.32$	$p$	$0.05$

Find the value of $p .$

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 6 - Probability Distributions>EXERCISE 6B>Q 2

The probability distribution for the random variable $Y$ is given in the following table.

$y$	$0$	$1$	$2$	$3$	$4$
$P (Y = y)$	$0.03$	$2 p$	$0.32$	$p$	$0.05$

Calculate $E (Y)$ and the standard deviation of $Y .$

A biased coin is tossed three times. The probability distribution for H, the number of heads obtained, is shown in the following table. h 0 1 2 3 P(H=h) 0.512 0.384 0.096 a Find the probability of obtaining a head each time the coin is tossed.

Important Questions on Probability Distributions

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A biased coin is tossed three times. The probability distribution for $H,$ the number of heads obtained, is shown in the following table.

$h$ $0$ $1$ $2$ $3$

$P (H = h)$ $0.512$ $0.384$ $0.096$ $a$

Find the probability of obtaining a head each time the coin is tossed.