calculate each of the following probabilities.\na. the probability that a nine - sided fair die will produce…

calculate each of the following probabilities.\na. the probability that a nine - sided fair die will produce an even number.\nb. a random digit from 1 to 9 (inclusive) is chosen, with all digits being equiprobable. the probability that its square will end in 1.\nc. the probability that a letter chosen from the alphabet (all letters being equiprobable) will be a vowel.\nd. a random number between 1 and 20 (exclusive) is chosen. the probability that its square root will not be an integer.

calculate each of the following probabilities.\na. the probability that a nine - sided fair die will produce an even number.\nb. a random digit from 1 to 9 (inclusive) is chosen, with all digits being equiprobable. the probability that its square will end in 1.\nc. the probability that a letter chosen from the alphabet (all letters being equiprobable) will be a vowel.\nd. a random number between 1 and 20 (exclusive) is chosen. the probability that its square root will not be an integer.

Answer

Explanation:

Step1: Calculate probability for A

A nine - sided fair die has 9 possible outcomes. The even numbers on it are 2, 4, 6, 8. So there are 4 favorable outcomes. The probability formula is $P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. So $P(A)=\frac{4}{9}$.

Step2: Calculate probability for B

The digits from 1 to 9 are considered. The squares of 1 is 1, 9 is 81. So 2 out of 9 digits have squares that end in 1. Using the probability formula, $P(B)=\frac{2}{9}$.

Step3: Calculate probability for C

There are 26 letters in the alphabet and 5 vowels (a, e, i, o, u). So $P(C)=\frac{5}{26}$.

Step4: Calculate probability for D

The numbers between 1 and 20 (exclusive) are 2 - 19. The perfect - square numbers in this range are 4, 9, 16. So there are 3 perfect - square numbers. Then there are 17 non - perfect - square numbers. So $P(D)=\frac{17}{18}$.

Answer:

A. $\frac{4}{9}$ B. $\frac{2}{9}$ C. $\frac{5}{26}$ D. $\frac{17}{18}$