4 tries, probability of success 0.5, probability of exactly 2… Step by Step

Step by step

The question

Question: There are 4 tries.

Each one is a success with probability 0.5.

What is the probability of exactly 2 successes?

1 Binomial: yes or no

4 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.

4 tries: same chance, each on its own

n is the number of tries: 4.

k is how many successes we want: 2.

p is the chance of a success each time: 0.5.

q is the chance of not: 1 minus p, 0.5.

n = 4 tries, k = 2 successes

p = 0.5 (success)

q = 1 − p = 0.5

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) = ₄C₂ × (0.5)² × (0.5)²

3 Count the orders

4 C 2 counts the orders: which 2 of the 4 tries are the successes.

4 times 3, over 2 times 1, is 6.

₄C₂ = 4 × 32 × 1 = 6

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

6 orders.

SSFF SFSF SFFS

FSSF FSFS FFSS

4 One order's chance

One order, like SSFF.

Along one path we multiply: 0.5 for each success, and 0.5 for each failure.

one order = 0.25 × 0.25

0.5² = 0.25

0.25 times 0.25 is 0.0625.

one order = 0.0625

5 Multiply

Every order has this same chance.

There are 6 orders, so we multiply by 6.

P(2 successes) = 6 × 0.0625

Careful!

That is only one order.

Successes can come in 6 different orders.

Careful: not 0.0625 ✗

6 times 0.0625 is 0.375.

P(2 successes) = 0.375

6 Check

Let's check another order, FFSS.

The same number of each, just in other places.

Multiplied, it is 0.0625 again.

So the answer is right!

FFSS: 0.5 × 0.5 × 0.5 × 0.5 = 0.0625

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