A die is rolled. Find the probability of the given event. Write your answers as whole numbers or reduced fractions.

(a) The number showing is a 5.
[tex]\[ P(5) = \square \][/tex]

(b) The number showing is an even number.
[tex]\[ P(\text{even}) = \square \][/tex]

(c) The number showing is greater than 4.
[tex]\[ P(\text{greater than 4}) = \square \][/tex]



Answer :

Sure, let's break down each part of the problem and find the probabilities step-by-step:

### (a) The number showing is a 5

A standard die has six faces, numbered from 1 to 6. Only one of these faces shows the number 5.

[tex]\[ P(5) = \frac{\text{Number of outcomes showing a 5}}{\text{Total number of outcomes}} \][/tex]

Since there is only one face with a 5:

[tex]\[ P(5) = \frac{1}{6} \][/tex]

So, the probability that the number showing is a 5 is:
[tex]\[ P(5) = \frac{1}{6} \][/tex]

### (b) The number showing is an even number

The even numbers on a die are 2, 4, and 6. There are three even numbers out of six total possible outcomes.

[tex]\[ P(\text{even}) = \frac{\text{Number of even outcomes}}{\text{Total number of outcomes}} \][/tex]

There are 3 even numbers (2, 4, 6) and 6 total numbers:

[tex]\[ P(\text{even}) = \frac{3}{6} \][/tex]

Simplifying this fraction:

[tex]\[ P(\text{even}) = \frac{1}{2} \][/tex]

So, the probability that the number showing is an even number is:
[tex]\[ P(\text{even}) = \frac{1}{2} \][/tex]

### (c) The number showing is greater than 4

The numbers greater than 4 on a die are 5 and 6. There are two such numbers out of six total possible outcomes.

[tex]\[ P(\text{greater than 4}) = \frac{\text{Number of outcomes greater than 4}}{\text{Total number of outcomes}} \][/tex]

There are 2 numbers greater than 4 (5, 6) and 6 total numbers:

[tex]\[ P(\text{greater than 4}) = \frac{2}{6} \][/tex]

Simplifying this fraction:

[tex]\[ P(\text{greater than 4}) = \frac{1}{3} \][/tex]

So, the probability that the number showing is greater than 4 is:
[tex]\[ P(\text{greater than 4}) = \frac{1}{3} \][/tex]

### Summary:

[tex]\[ \begin{aligned} (a) & \quad P(5) = \frac{1}{6} \\ (b) & \quad P(\text{even}) = \frac{1}{2} \\ (c) & \quad P(\text{greater than 4}) = \frac{1}{3} \\ \end{aligned} \][/tex]