Roll a die 5 times, probability of exactly 1 one Step by Step

Step by step

The question

Question: A die is rolled 5 times.

What is the probability of exactly 1 one?

1 Binomial: yes or no

5 rolls.

Each one has the same chance, and one does not change the next.

Each is a success or not.

That is a binomial experiment.

5 rolls: same chance, each on its own

n is the number of rolls: 5.

k is how many ones we want: 1.

p is the chance of a success each time: 1/6.

q is the chance of not: 1 minus p, 5/6.

n = 5 rolls, k = 1 ones

p = 16 (on)

q = 1 − p = 56

2 Write the rule

The rule: n C k, times p to the power k, times q to the power n minus k.

P(1 one) = _nC_k × p^k × q^{n − k}

We put the numbers in.

P(1 one) = ₅C₁ × (16)¹ × (56)⁴

3 Count the orders

5 C 1 counts the orders: the one one can be any of the 5 rolls.

That is 5.

₅C₁ = 51 = 5

Here they are: 1 is a one, N is a roll that is not a 1.

5 orders.

1NNNN N1NNN NN1NN NNN1N NNNN1

4 One order's chance

One order, like 1NNNN.

Along one path we multiply: 1/6 for each one, and 5/6 for each roll that is not a 1.

one order = 16 × 6251296

(56)⁴ = 6251296

Top times top, bottom times bottom: 625/7776.

one order = 6257776

5 Multiply

Every order has this same chance.

There are 5 orders, so we multiply by 5.

P(1 one) = 5 × 6257776

Careful!

That is only one order.

Ones can come in 5 different orders.

Careful: not 6257776 ✗

5 times 625 is 3125.

So it is 3125/7776.

P(1 one) = 31257776

As a decimal, to 4 places, that is about 0.4019.

P(1 one) ≈ 0.4019

6 Check

Let's check another order, NNNN1.

The same number of each, just in other places.

Multiplied, it is 625/7776 again.

So the answer is right!

NNNN1: 56 × 56 × 56 × 56 × 16 = 6257776

Let's look back.

First: binomial: yes or no.

Then: write the rule.

Then: count the orders.

Then: one order's chance.

Then: multiply.

Last: check.

Answer