Question

A fair die is rolled seven times. What is the probability that it comes up 5 at least once?

Answer 1

The probability of rolling a 5 at least once in seven rolls of a fair die is approximately 0.665.

Step-by-step explanation:

To find the probability of rolling a 5 at least once in seven rolls of a fair die, we can calculate the probability of the complementary event, which is rolling a 5 zero times in seven rolls.

The probability of not rolling a 5 in a single roll is 5/6, since there are five numbers that are not 5 on the die.

The probability of not rolling a 5 in seven rolls is (5/6)^7. The probability of rolling a 5 at least once in seven rolls is therefore 1 - (5/6)^7, which is approximately 0.665.

Answer 2

The probability of rolling a 5 at least once when rolling a fair die seven times is approximately 0.665.

Step-by-step explanation:

To find the probability of rolling a 5 at least once when rolling a fair die seven times, we can use the complement rule. The complement rule states that the probability of an event not happening is equal to 1 minus the probability of the event happening. In this case, the event is not rolling a 5 at all. The probability of rolling a number that is not 5 on any given roll is 5/6, since there are 6 numbers on the die and only 1 of them is a 5. To find the probability of not rolling a 5 over seven rolls, we can multiply the probabilities of not rolling a 5 on each roll:

1. P(not rolling a 5) = 5/6
2. P(not rolling a 5 on two rolls) = (5/6) * (5/6)
3. P(not rolling a 5 on three rolls) = (5/6) * (5/6) * (5/6)
4. ...
5. P(not rolling a 5 on seven rolls) = (5/6) * (5/6) * (5/6) * (5/6) * (5/6) * (5/6) * (5/6)

The probability of rolling a 5 at least once is then equal to 1 minus the probability of not rolling a 5 on any of the seven rolls:

P(rolling a 5 at least once) = 1 - P(not rolling a 5 on seven rolls)

By plugging in the formula for P(not rolling a 5 on seven rolls), we can calculate the final probability:

P(rolling a 5 at least once) = 1 - (5/6)7 ≈ 0.665