Question

If a fair 5-sided die is rolled 5 times, what is the probability that each possible outcome (1, 2, 3, 4, and 5) will occur exactly once

Answer 1

0.0384

Explanation:

Given a fair 5 sided die rolled 5 times.

Numbers on it are 1, 2, 3, 4, 5.

To find:

The probability that each number will occur exactly once.

Solution:

Formula for probability of an event E:

`P(E) = \frac{\text{Number of favorable cases}}{\text {Total number of cases}}`

At the first roll of die, any number can occur.

So number of possible outcomes = 5

Total number of possible outcomes = 5

`P(1^(st)\ roll) = (5)/(5) = 1`

At the second roll of die, any number can occur other than that occurred in first roll.

So number of possible outcomes = 4

Total number of possible outcomes = 5

`P(2^(nd)\ roll) = (4)/(5)`

At the third roll of die, any number can occur other than that occurred in first and second roll.

So number of possible outcomes = 3

Total number of possible outcomes = 5

`P(3^(rd)\ roll) = (3)/(5)`

At the fourth roll of die, any number can occur other than that occurred in first, second and third roll.

So number of possible outcomes = 2

Total number of possible outcomes = 5

`P(4^(th)\ roll) = (2)/(5)`

At the fifth roll of die, any number can occur other than that occurred in first, second, third and fourth roll.

So number of possible outcomes = 1

Total number of possible outcomes = 5

`P(5^(th)\ roll) = (1)/(5)`

The required probability will be multiplication of all the five probabilities.

`1 * 0.8 * 0.6 * 0.4 * 0.2 = \bold{0.0384}`