Probability of exactly 4 heads in 4 coin flips Step by Step

Step by step

The question

Question: A coin is flipped 4 times.

What is the probability of exactly 4 heads?

1 Binomial: yes or no

4 flips.

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

Each is a success or not.

That is a binomial experiment.

4 flips: same chance, each on its own

n is the number of flips: 4.

k is how many heads we want: 4.

p is the chance of a success each time: 1/2.

q is the chance of not: 1 minus p, 1/2.

n = 4 flips, k = 4 heads

p = 12 (head)

q = 1 − p = 12

2 Write the rule

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

P(4 heads) = _nC_k × p^k × q^{n − k}

We put the numbers in.

P(4 heads) = ₄C₄ × (12)⁴ × (12)⁰

3 Count the orders

4 C 4 counts the orders: which 4 of the 4 flips are the heads.

4 times 3 times 2 times 1, over 4 times 3 times 2 times 1, is 1.

₄C₄ = 4 × 3 × 2 × 14 × 3 × 2 × 1 = 1

4 One order's chance

One order, like HHHH.

Along one path we multiply: 1/2 for each head.

one order = 116 × 1

(12)⁴ = 116

Top times top, bottom times bottom: 1/16.

one order = 116

5 Multiply

There is only 1 order: every one is a success.

Times 1.

P(4 heads) = 1 × 116

1 times 1 is 1.

So it is 1/16.

P(4 heads) = 116

6 Check

And count it: 4 flips make 16 outcomes, all just as likely.

1 of them have exactly 4 heads.

That is 1/16.

So the answer is right!

1 of 2 × 2 × 2 × 2 = 16 outcomes: 116

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