5 throws, probability of success 0.2, probability of exactly 1 hit Step by Step

Step by step

The question

Question: There are 5 throws.

Each one is a success with probability 0.2.

What is the probability of exactly 1 success?

1 Binomial: yes or no

5 throws.

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 throws: same chance, each on its own

n is the number of throws: 5.

k is how many successes we want: 1.

p is the chance of a success each time: 0.2.

q is the chance of not: 1 minus p, 0.8.

n = 5 throws, k = 1 successes

p = 0.2 (success)

q = 1 − p = 0.8

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 succes) = _nC_k × p^k × q^{n − k}

We put the numbers in.

P(1 succes) = ₅C₁ × (0.2)¹ × (0.8)⁴

3 Count the orders

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

That is 5.

₅C₁ = 51 = 5

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

5 orders.

SFFFF FSFFF FFSFF FFFSF FFFFS

4 One order's chance

One order, like SFFFF.

Along one path we multiply: 0.2 for each success, and 0.8 for each failure.

one order = 0.2 × 0.4096

0.8⁴ = 0.4096

0.2 times 0.4096 is 0.08192.

one order = 0.08192

5 Multiply

Every order has this same chance.

There are 5 orders, so we multiply by 5.

P(1 succes) = 5 × 0.08192

Careful!

That is only one order.

Successes can come in 5 different orders.

Careful: not 0.08192 ✗

5 times 0.08192 is 0.4096.

P(1 succes) = 0.4096

6 Check

Let's check another order, FFFFS.

The same number of each, just in other places.

Multiplied, it is 0.08192 again.

So the answer is right!

FFFFS: 0.8 × 0.8 × 0.8 × 0.8 × 0.2 = 0.08192

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