A fair die is...

Question

A fair die is thrown 7 times. What is the probability of getting point 1 on at most of the 7 throws?

Answer 1

Interpreting your question as:

A fair die is thrown 7 times. What is the probability of getting a 1 on at most one of the throws?

First, let’s state a few standard assumptions:

1. A fair die has 6 equally likely sides, numbered 1 through 6.
2. The throws (also called trials) are independent, meaning the outcome of one throw does not affect the others.
3. “At most one” means either zero or exactly one throw comes up as a 1.

The probability of getting a specific outcome (such as a 1) on a single throw is ( 1/6 ), and the probability of not getting a 1 on a throw is ( 5/6 ).

We’ll also use two basic probability rules:

1. The probability of one of several mutually exclusive events occurring is the sum of their probabilities.
2. The probability of independent events occurring together is the product of their probabilities.

Now consider the possible ways the success condition can occur. These events are mutually exclusive:

1. Zero of the 7 throws come up as 1
2. Exactly one throw comes up as 1:
3. the first throw is a 1 and the other 6 are not
4. the second throw is a 1 and the other 6 are not
5. …
6. the seventh throw is a 1 and the other 6 are not

We can write this compactly as:

(5/6)⁷ + 7•(1/6)(5/6)⁶

Now simplify:

(5/6)⁷ + 7•(1/6)(5/6)⁶

= 5⁷/6⁷ + (7•5⁶)/6⁷

= (5⁷ + 7•5⁶)/6⁷

= (5⁶(5+7))/6⁷

= (5⁶•12)/6⁷

= (5⁶•2)/6⁶

= (5⁶)/(6⁵•3)

= 15625/23328

Decimal approximation:

≈ 0.669796

So the probability is (15625/23328 ≈ 0.6698).