n whole numbers are randomly chosen and multiplied. Now, match the following lists.
`{:(" List I"," List II"),("a. The probability that the last digit is 1, 3, 7, or 9 is","p. "(8^(n)-4^(n))/(10^(n))),("b. The probability that the last digit is 2, 4, 6, 8 is ","q. "(5^(n) - 4^(n))/(10^(n))),("c. The probability that the last digit is 5 is","r. " (4^(n))/(10^(n))),("d. The probability that the last digit is zero is","s. "(10^(n) - 8^(n) - 5^(n) + 4^(n))/(10^(n))):}`

Options

A. `{:("a","b","c","d"),("q","s","s","r"):}`
B. `{:("a","b","c","d"),("r","q","q","p"):}`
C. `{:("a","b","c","d"),("q","p","p","s"):}`
D. `{:("a","b","c","d"),("q","s","p","r"):}`

Answer

A, B, C, D

Explanation

a. The required event will occur if last digit in all the chosen numbers is 1, 3, 7, or 9. Therefore, the required probability is `(4//10)^(n)`.
b. The required probability is equal to the probability that the last digit is 2, 4, 6, 8 and is given by
P(last digit is 1, 2, 3, 4, 6, 7, 8, 9)
-P (last digit is 1, 3, 7, 9) = `(8^(n) - 4^(n))/(10^(n))`
c. P(1, 3, 5, 7, 9) - P(1, 3, 7, 9) = `(5^(n) - 4^(n))/(10^(n))`
d. The required probability is
`P(0, 5) - P(5) = ((10^(n) - 8^(n))-(5^(n) - 4^(n)))/(10^(n))`
`P(0, 5) - P(5) = ((10^(n) - 8^(n))-(5^(n) - 4^(n)))/(10^(n))`