A basketball player makes 60% of free throws, independently. In 5… Step by Step

Step by step

The question

Question: There are 5 tries.

Each one is a success with probability 0.6.

What is the probability of exactly 3 successes?

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Binomial: yes or no

5 tries.

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 tries: same chance, each on its own

n is the number of tries: 5.

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 = 5 tries

k = 3 successes

p = 0.6 (success)

q = 1 − p

= 0.4

2 Write the rule

The rule has two parts.

n C k counts the orders.

p to the power k, times q to the power n minus k, is the chance of one order.

We build each part, then multiply.

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

We put the numbers in.

P(3 successes) = ₅C₃ × (0.6)³ × (0.4)²

3 Count the orders

5 C 3 counts the orders: which 3 of the 5 tries are the successes.

5 times 4 times 3, over 3 times 2 times 1, is 10.

₅C₃ = 5 × 4 × 33 × 2 × 1 = 10

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

10 orders.

SSSFF SSFSF SSFFS SFSSF SFSFS

SFFSS FSSSF FSSFS FSFSS FFSSS

4 One order's chance

One order, like SSSFF.

For one order we multiply the chances, one for each try, as in the and lessons: 0.6 for each success, and 0.4 for each failure.

one order = 0.216 × 0.16

0.6³ = 0.216

0.4² = 0.16

0.216 times 0.16 is 0.03456.

one order = 0.03456

5 Multiply

Every order has this same chance.

There are 10 orders, so we multiply by 10.

P(3 successes) = 10 × 0.03456

Careful!

That is only one order.

Successes can come in 10 different orders.

Careful: not 0.03456 ✗

10 times 0.03456 is 0.3456.

P(3 successes) = 0.3456

6 Check

Let's check another order, FFSSS.

The same number of each, just in other places.

Multiplied, it is 0.03456 again.

So the answer is right!

FFSSS: 0.4 × 0.4 × 0.6 × 0.6 × 0.6 = 0.03456

7 Pick the choice

Our answer is P(3 successes) = 0.3456.

That is choice C.

C) 0.3456

8 Why not the others

A is 0.216.

Wrong: 0.6³ only: the 2 misses and the 10 orders count too.

A) 0.216 ✗ 0.6^3 only: the 2 misses and the 10 orders count too

B is 0.6.

Wrong: the chance of one throw.

B) 0.6 ✗ the chance of one throw

D is 0.03456.

Wrong: one order only: there are 10 orders.

That is the Careful mistake from before.

D) 0.03456 ✗ one order only: there are 10 orders

Let's look back.

First: binomial: yes or no.

Then: write the rule.

Then: count the orders.

Then: one order's chance.

Then: multiply.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer