Roll a die 3 times, probability of exactly 2 sixes Step by Step

Step by step

The question

Question: A die is rolled 3 times.

What is the probability of exactly 2 sixes?

1 Binomial: yes or no

3 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.

3 rolls: same chance, each on its own

n is the number of rolls: 3.

k is how many sixes we want: 2.

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

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

n = 3 rolls, k = 2 sixes

p = 16 (six)

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(2 sixes) = _nC_k × p^k × q^{n − k}

We put the numbers in.

P(2 sixes) = ₃C₂ × (16)² × (56)¹

3 Count the orders

3 C 2 counts the orders: which 2 of the 3 rolls are the sixes.

3 times 2, over 2 times 1, is 3.

₃C₂ = 3 × 22 × 1 = 3

Here they are: 6 is a six, N is a roll that is not a 6.

3 orders.

66N 6N6 N66

4 One order's chance

One order, like 66N.

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

one order = 136 × 56

(16)² = 136

Top times top, bottom times bottom: 5/216.

one order = 5216

5 Multiply

Every order has this same chance.

There are 3 orders, so we multiply by 3.

P(2 sixes) = 3 × 5216

Careful!

That is only one order.

Sixes can come in 3 different orders.

Careful: not 5216 ✗

3 times 5 is 15.

So it is 15/216.

P(2 sixes) = 15216

Both 15 and 216 can be divided by 3.

That makes it simpler.

P(2 sixes) = 15 ÷ 3216 ÷ 3

So it is 5/72.

P(2 sixes) = 572

6 Check

Let's check another order, N66.

The same number of each, just in other places.

Multiplied, it is 5/216 again.

So the answer is right!

N66: 56 × 16 × 16 = 5216

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