A fair 4 -sided spinner with sides labelled A , A , B , B is spun four times. By drawing up the probability distribution table for X , the number of times the spinner comes to rest on B , find the value of Var X E X .

Given that, Arrangement or list A A B B , A B A B , A B B A , B A A B , B A B A and B B A A Equally likely because each has a probability equal to the product of 1 4 , 1 4 , 3 4 and 3 4 P ( X = 0 ) = 1 4 × 1 4 × 1 4 × 1 4 = 1 256 P ( X = 2 ) = 6 × 3 4 × 3 4 × 1 4 × 1 4 = 54 256 P ( X = 4 ) = 3 4 × 3 4 × 3 4 × 3 4 = 81 256 P ( X = 1 ) = 4 × 3 4 × 1 4 × 1 4 × 1 4 = 12 256 P ( X = 3 ) = 4 × 3 4 × 3 4 × 3 4 × 1 4 = 108 256 x 0 1 2 3 4 P X = x 1 256 12 256 54 256 108 256 81 256 E ( X ) = 1 256 × ( 0 + 12 + 108 + 324 + 324 ) = 3 Var ( X ) = 1 256 × 0 2 × 1 + 1 2 × 12 + 2 2 × 54 + 3 2 × 108 + 4 2 × 81 - 3 2 = 3 4 Var ( X ) E ( X ) = 3 4 ÷ 3 = 1 4 1 4 represents the probability of not obtaining B with each spin.

The following table shows the probability distribution for the random variable Y . y 1 10 q 101 P ( Y = y ) 0 . 2 0 . 4 0 . 2 0 . 2 Given that Var Y = 1385 . 2 , show that q 2 - 61 q + 624 = 0 and solve this equation.

Given that, y 1 10 q 101 P ( Y = y ) 0 . 2 0 . 4 0 . 2 0 . 2 We know that, V a r ( Y ) = E Y 2 - [ E ( Y ) ] 2 = ∑ i = 1 n p i y i 2 - ∑ i = 1 n p i y i 2 = ( 1 ) 2 × 0 . 1 + ( 10 ) 2 × 0 . 4 + ( 9 ) 2 × 0 . 2 + ( 101 ) 2 × 0 . 2 - [ 1 × 0 . 2 + 10 × 0 . 4 + 9 × 0 . 2 + 101 × 0 . 2 ] 2 ⇒ 1385 · 2 = 0 . 2 + 40 + 0 . 2 q 2 + 2040 . 2 - [ 0 . 2 + 4 + 0 . 2 q + 20 · 2 ] 2 ⇒ 1385 . 2 = 0 . 2 q 2 + 2080 · 4 - 0 . 04 q 2 - 9 . 76 q - 595 . 36 ∴ 1385 . 2 = 0 . 16 q 2 - 9 . 762 + 1485 . 04 ∴ 0 . 16 q 2 - 9 . 76 q + 99 . 84 = 0 ⇒ q 2 - 619 + 624 = 0 ⇒ q = 48 , 13

The following table shows the probability distribution for the random variable Y . y 1 10 q 101 P ( Y = y ) 0 . 2 0 . 4 0 . 2 0 . 2 Find the greatest possible value of E Y if the Var Y = 1385 . 2 .

Given that, y 1 10 q 101 P ( Y = y ) 0 . 2 0 . 4 0 . 2 0 . 2 We know that, Var ( Y ) = E Y 2 - [ E ( Y ) ] 2 Var ( Y ) = ∑ i = 1 n p i y i 2 - ∑ t = 1 n p i y i 2 = ( 1 ) 2 × 0 . 2 + ( 10 ) 2 × 0 . 4 + ( q ) 2 × 0 . 2 + ( 101 ) 2 × 0 . 2 - [ 1 × 0 . 2 + 10 × 0 . 4 + q × 0 . 2 + 101 × 0 . 2 ] 2 ⇒ 1385 · 2 = 0 . 2 + 40 + 0 . 2 q 2 + 2040 . 2 - [ 0 . 2 + 4 + 0 . 2 q + 20 · 2 ] 2 ⇒ 1385 . 2 = 0 . 2 q 2 + 2080 · 4 - 0 . 04 q 2 - 9 . 76 q - 595 . 36 ∴ 1385 . 2 = 0 . 16 q 2 - 9 . 762 + 1485 . 04 ∴ 0 . 16 q 2 - 9 . 76 q + 99 . 84 = 0 ⇒ q 2 - 619 + 624 = 0 ⇒ q = 48 & q = 13 at q = 13 E ( Y ) = ∑ i = 1 n p i y i = 1 × 0 . 2 + 10 × 0 . 4 + 13 × 0 . 2 + 0 . 2 × 101 = 0 . 2 + 4 + 2 . 6 + 20 · 2 ⇒ E Y = 27 at q = 48 , ⇒ E ( Y ) = ∑ i = 1 n P i y i = 1 × 0 . 2 + 10 × 0 . 4 + 48 × 0 . 2 + 0 . 2 × 101 = 0 . 2 + 20 . 2 + 4 + 9 . 6 = 34 Greatest possible value of E ( y ) = 34 maximum value of ( 27 , 34 ) .

An investment company has produced the following table, which shows the probabilities of various percentage profits on money invested over a period of 3 years. Profit % 1 5 10 15 20 30 40 45 50 Probability 0 . 05 0 . 10 0 . 50 0 . 20 0 . 05 0 . 04 0 . 03 0 . 02 0 . 01 Calculate the expected profit on an investment of $ 50000 .

Given that, An investment company has produced the following table, which shows the probabilities of various percentage profits on money invested over a period of 3 years. Profit % 1 5 10 15 20 30 40 45 50 Probability 0 . 05 0 . 10 0 . 50 0 . 20 0 . 05 0 . 04 0 . 03 0 . 02 0 . 01 Expected profit = ∑ i = 1 n x i p i = 1 × 0 . 05 + 5 × 0 . 10 + 15 × 0 . 2 + 20 × 0 . 05 + 30 × 0 . 04 + 40 × 0 . 03 + 45 × 0 . 02 + 50 × 0 . 01 + 10 × 0 . 50 = 13 . 35 % ∴ Expected profit on investment $ 50000 = 50000 × 13 . 35 100 = $ 6675 ∴ Expected profit = 13 . 35 % × 50 , 000 = $ 6675

An investment company has produced the following table, which shows the probabilities of various percentage profits on money invested over a period of 3 years. Profit % 1 5 10 15 20 30 40 45 50 Probability 0 . 05 0 . 10 0 . 50 0 . 20 0 . 05 0 . 04 0 . 03 0 . 02 0 . 01 A woman considers investing $ 50000 with the company, but decides that her money is likely to earn more when invested over the same period in a savings account that pays r % compound interest per annum. Calculate, correct to 2 decimal places, the least possible value of r .

Given that, An investment company has produced the following table, which shows the probabilities of various percentage profits on money invested over a period of 3 years. Profit % 1 5 10 15 20 30 40 45 50 Probability 0 . 05 0 . 10 0 . 50 0 . 20 0 . 05 0 . 04 0 . 03 0 . 02 0 . 01 Expected profit = ∑ i = 1 n x i p i = 1 × 0 . 05 + 5 × 0 . 10 + 15 × 0 . 2 + 20 × 0 . 05 + 30 × 0 . 04 + 40 × 0 . 03 + 45 × 0 . 02 + 50 × 0 . 01 + 10 × 0 . 50 = 13 . 35 % ∴ Expected profit on investment $ 50000 = 50000 × 13 . 35 100 = $ 6675 ∴ Expected profit = 13 . 35 % × 50 , 000 = $ 6675 In compound interest, Amount = P 1 + r 100 T p = Principle = $ 50 , 000 , T = 3 , r = ? A = 50 , 020 + 6675 = 56675 ⇒ 50000 × 1 + r 10 3 > 56675 ∴ r = 4 . 27

EASY

AS and A Level

IMPORTANT

Earn 100

An ordinary fair die is rolled. If the die shows an odd number then $S,$ the score awarded, is equal to that number. If the die shows an even number, then the die is rolled again. If on the second roll it shows an odd number, then that is the score awarded. If the die shows an even number on the second roll, the score awarded is equal to half of that even number.

Calculate the exact value of Var $(S)$

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 6B>Q 15

A fair $4$ -sided spinner with sides labelled $A, A, B, B$ is spun four times.

Show that there are six equally likely ways to obtain exactly two $B s$ with the four spins.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 6 - Probability Distributions>EXERCISE 6B>Q 15

A fair $4$ -sided spinner with sides labelled $A, A, B, B$ is spun four times.

By drawing up the probability distribution table for $V ar (s) = E (s^{2}) - [E (s)]^{2} = \sum_{i = 1}^{n} P_{i} {S_{i}}^{2} - {[\sum_{i = 1}^{n} p_{i} x_{i}]}^{2}$ the number of times the spinner comes to rest on $B,$ find the value of $\frac{Var (X)}{E (X)} .$

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 6 - Probability Distributions>EXERCISE 6B>Q 15

A fair $4$ -sided spinner with sides labelled $A, B, B, B$ is spun four times.

What, in the context of this question, does the value found in part $b$ represent?

EASY

AS and A Level

IMPORTANT

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

Find the mean and the variance of the discrete random variable $X,$ whose probability distribution is given in the following table.

$x$	$1$ .	$2$	$3$	$4$
$P (X = x)$	$1 - k$	$2 - 3 k$	$3 - 4 k$	$4 - 6 k$

EASY

AS and A Level

IMPORTANT

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

The following table shows the probability distribution for the random variable $Y .$

$y$	$1$	$10$	$q$	$101$
$P (Y = y)$	$0.2$	$0.4$	$0.2$	$0.2$

Given that Var $(Y) = 1385.2,$ show that $q^{2} - 61 q + 624 = 0$ and solve this equation.

EASY

AS and A Level

IMPORTANT

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

The following table shows the probability distribution for the random variable $Y .$

$y$	$1$	$10$	$q$	$101$
$P (Y = y)$	$0.2$	$0.4$	$0.2$	$0.2$

Find the greatest possible value of $E (Y)$ if the $Var (Y) = 1385.2$ .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 6 - Probability Distributions>END-OF-CHAPTER REVIEW EXERCISE 6>Q 3

An investment company has produced the following table, which shows the probabilities of various percentage profits on money invested over a period of $3$ years.

Profit $(%)$	$1$	$5$	$10$	$15$	$20$	$30$	$40$	$45$	$50$
Probability	$0.05$	$0.10$	$0.50$	$0.20$	$0.05$	$0.04$	$0.03$	$0.02$	$0.01$

Calculate the expected profit on an investment of $$ 50000 .$

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 6 - Probability Distributions>END-OF-CHAPTER REVIEW EXERCISE 6>Q 3

An investment company has produced the following table, which shows the probabilities of various percentage profits on money invested over a period of $3$ years.

Profit $(%)$	$1$	$5$	$10$	$15$	$20$	$30$	$40$	$45$	$50$
Probability	$0.05$	$0.10$	$0.50$	$0.20$	$0.05$	$0.04$	$0.03$	$0.02$	$0.01$

A woman considers investing $$ 50000$ with the company, but decides that her money is likely to earn more when invested over the same period in a savings account that pays $r %$ compound interest per annum. Calculate, correct to $2$ decimal places, the least possible value of $r .$

Important Questions on Probability Distributions

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