In a state lottery, four digits are drawn at random one at a time with replacement from 0 to 9. Suppose that you win if any permutation of your selected inte- gers is drawn. Give the probability of winning if you select

In a state lottery, four digits are drawn at random one at a time with replacement from 0 to 9. Suppose that you win if any permutation of your selected integers is drawn. Give the probability of winning if you select a. 6, 7, 8, 9. b. 6, 7, 8, 8. c. 7, 7, 8, 8. d. 7, 8, 8, 8.

a. Since there are 10 digits (0 to 9), the number of different ways of obtaining a four digit number is 10 × 10 × 10 × 10 = 10000 ways.

Now, the number of digits that can be formed from 6,7,8,9 are (since they are 4 digits and order is important) ⁴P₄ = 4! = 24 ways

So, the probability of obtaining 6, 7, 8 ,9 is

Since probability = number of desired outcome/number of possible outcomes

Since our number of desired outcome = number of different ways of obtaining 6, 7, 8 ,9 = 24 ways and number of possible outcomes = 10000 ways

P = 24/10000 = 0.0024

b. Since there are 10 digits (0 to 9), the number of different ways of obtaining a four digit number is 10 × 10 × 10 × 10 = 10000 ways.

Now, the number of digits that can be formed from 6,7,8,8 are (since they are 4 digits and order is important) ⁴P₄ = 4!. Since we have 2 eights, we divide by 2!. So 4!/2! = 24/2 ways = 12 ways

So, the probability of obtaining 6, 7, 8 ,8 is

Since probability = number of desired outcome/number of possible outcomes

Since our number of desired outcome = number of different ways of obtaining 6, 7, 8 ,8 = 12 ways and number of possible outcomes = 10000 ways

P = 12/10000 = 0.0012

c. Since there are 10 digits (0 to 9), the number of different ways of obtaining a four digit number is 10 × 10 × 10 × 10 = 10000 ways.

Now, the number of digits that can be formed from 7,7,8,8 are (since they are 4 digits and order is important) ⁴P₄ = 4!. Since we have 2 seven's and 2 eights , we divide by 2!2!. So 4!/2!2! = 24/4 ways = 6 ways

So, the probability of obtaining 7, 7, 8 ,8 is

Since probability = number of desired outcome/number of possible outcomes

Since our number of desired outcome = number of different ways of obtaining 7, 7, 8 ,8 = 6 ways and number of possible outcomes = 10000 ways

P = 6/10000 = 0.0006

d. Since there are 10 digits (0 to 9), the number of different ways of obtaining a four digit number is 10 × 10 × 10 × 10 = 10000 ways.

Now, the number of digits that can be formed from 7,8,8,8 are (since they are 4 digits and order is important) ⁴P₄ = 4!. Since we have 3 eights , we divide by 3!. So 4!/3! = 24/6 ways = 4 ways

So, the probability of obtaining 7, 8, 8 ,8 is

Since probability = number of desired outcome/number of possible outcomes

Since our number of desired outcome = number of different ways of obtaining 7, 8, 8 ,8 = 4 ways and number of possible outcomes = 10000 ways

P = 4/10000 = 0.0004