5 tries, probability of success 0.3, probability of exactly 2… Step by Step

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The question

Question: There are 5 tries.

Each one is a success with probability 0.3.

What is the probability of exactly 2 successes?

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: 2.

p is the chance of a success each time: 0.3.

q is the chance of not: 1 minus p, 0.7.

n = 5 tries, k = 2 successes

p = 0.3 (success)

q = 1 − p = 0.7

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 successes) = _nC_k × p^k × q^{n − k}

We put the numbers in.

P(2 successes) = ₅C₂ × (0.3)² × (0.7)³

3 Count the orders

5 C 2 counts the orders: which 2 of the 5 tries are the successes.

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

₅C₂ = 5 × 42 × 1 = 10

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

10 orders.

SSFFF SFSFF SFFSF SFFFS FSSFF

FSFSF FSFFS FFSSF FFSFS FFFSS

4 One order's chance

One order, like SSFFF.

Along one path we multiply: 0.3 for each success, and 0.7 for each failure.

one order = 0.09 × 0.343

0.3² = 0.09

0.7³ = 0.343

0.09 times 0.343 is 0.03087.

one order = 0.03087

5 Multiply

Every order has this same chance.

There are 10 orders, so we multiply by 10.

P(2 successes) = 10 × 0.03087

Careful!

That is only one order.

Successes can come in 10 different orders.

Careful: not 0.03087 ✗

10 times 0.03087 is 0.3087.

P(2 successes) = 0.3087

6 Check

Let's check another order, FFFSS.

The same number of each, just in other places.

Multiplied, it is 0.03087 again.

So the answer is right!

FFFSS: 0.7 × 0.7 × 0.7 × 0.3 × 0.3 = 0.03087

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