Step by step
The question
Question: There are 6 games.
Each one is a success with probability 40%.
What is the probability of exactly 4 successes?
1 Binomial: yes or no
6 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: 6.
k is how many successes we want: 4.
p is the chance of a success each time: 0.4.
q is the chance of not: 1 minus p, 0.6.
2 Write the rule
The rule: n C k, times p to the power k, times q to the power n minus k.
We put the numbers in.
3 Count the orders
6 C 4 counts the orders: which 4 of the 6 games are the successes.
6 times 5 times 4 times 3, over 4 times 3 times 2 times 1, is 15.
4 One order's chance
One order, like this one.
Along one path we multiply: 0.4 for each success, and 0.6 for each failure.
0.0256 times 0.36 is 0.009216.
5 Multiply
Every order has this same chance.
There are 15 orders, so we multiply by 15.
Careful!
That is only one order.
Successes can come in 15 different orders.
15 times 0.009216 is 0.13824.
To 4 decimal places: about 0.1382.
6 Check
Let's check: 15 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.