3 seeds, probability of success 0.8, probability of exactly 3 grow Step by Step

Step by step

The question

Question: There are 3 seeds.

Each one is a success with probability 0.8.

What is the probability of exactly 3 successes?

1 Binomial: yes or no

3 seeds.

Each one has the same chance, and one does not change the next.

Each is a success or not.

That is a binomial experiment.

3 seeds: same chance, each on its own

n is the number of seeds: 3.

k is how many successes we want: 3.

p is the chance of a success each time: 0.8.

q is the chance of not: 1 minus p, 0.2.

n = 3 seeds, k = 3 successes

p = 0.8 (success)

q = 1 − p = 0.2

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) = ₃C₃ × (0.8)³ × (0.2)⁰

3 Count the orders

3 C 3 counts the orders: which 3 of the 3 seeds are the successes.

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

₃C₃ = 3 × 2 × 13 × 2 × 1 = 1

4 One order's chance

One order, like SSS.

Along one path we multiply: 0.8 for each success.

one order = 0.512 × 1

0.8³ = 0.512

0.512 times 1 is 0.512.

one order = 0.512

5 Multiply

There is only 1 order: every one is a success.

Times 1.

P(3 successes) = 1 × 0.512

1 times 0.512 is 0.512.

P(3 successes) = 0.512

6 Check

Let's check: 1 order, each with the same chance.

So the answer is right!

1 order × one order's chance

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