4 shots, probability of success 0.6, probability of exactly 3 hits Step by Step

Step by step

The question

Question: There are 4 shots.

Each one is a success with probability 0.6.

What is the probability of exactly 3 successes?

1 Binomial: yes or no

4 shots.

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 shots: same chance, each on its own

n is the number of shots: 4.

k is how many successes we want: 3.

p is the chance of a success each time: 0.6.

q is the chance of not: 1 minus p, 0.4.

n = 4 shots, k = 3 successes

p = 0.6 (success)

q = 1 − p = 0.4

2 Write the rule

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

P(3 successes) = _nC_k × p^k × q^{n − k}

We put the numbers in.

P(3 successes) = ₄C₃ × (0.6)³ × (0.4)¹

3 Count the orders

4 C 3 counts the orders: which 3 of the 4 shots are the successes.

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

₄C₃ = 4 × 3 × 23 × 2 × 1 = 4

Here they are: S is a success, F is a failure.

4 orders.

SSSF SSFS SFSS FSSS

4 One order's chance

One order, like SSSF.

Along one path we multiply: 0.6 for each success, and 0.4 for each failure.

one order = 0.216 × 0.4

0.6³ = 0.216

0.216 times 0.4 is 0.0864.

one order = 0.0864

5 Multiply

Every order has this same chance.

There are 4 orders, so we multiply by 4.

P(3 successes) = 4 × 0.0864

Careful!

That is only one order.

Successes can come in 4 different orders.

Careful: not 0.0864 ✗

4 times 0.0864 is 0.3456.

P(3 successes) = 0.3456

6 Check

Let's check another order, FSSS.

The same number of each, just in other places.

Multiplied, it is 0.0864 again.

So the answer is right!

FSSS: 0.4 × 0.6 × 0.6 × 0.6 = 0.0864

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