Step by step
The question
Question: There are 8 games.
Each one is a success with probability 30%.
What is the probability of exactly 2 successes?
1 Binomial: yes or no
8 games.
Each one has the same chance, and one does not change the next.
Each is a success or not.
That is a binomial experiment.
n is the number of games: 8.
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.
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.
We put the numbers in.
3 Count the orders
8 C 2 counts the orders: which 2 of the 8 games are the successes.
8 times 7, over 2 times 1, is 28.
4 One order's chance
One order, like this one.
For one order we multiply the chances, one for each try, as in the and lessons: 0.3 for each success, and 0.7 for each failure.
0.09 times 0.117649 is 0.01058841.
5 Multiply
Every order has this same chance.
There are 28 orders, so we multiply by 28.
Careful!
That is only one order.
Successes can come in 28 different orders.
28 times 0.01058841 is 0.29647548.
To 4 decimal places: about 0.2965.
6 Check
Let's check: 28 orders, each with the same chance.
So the answer is right!
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.