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?
There are four choices.
First we solve it, the normal way.
Then we pick.
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 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
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.
For one order we multiply the chances, one for each try, as in the and lessons: 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!
7 Pick the choice
Our answer is P(4 successes) ≈ 0.1382.
That is choice C.
8 Why not the others
A is 0.0256.
Wrong: 0.4⁴ only: there are 15 orders and 2 losses.
B is 0.4.
Wrong: the chance of one win.
D is 0.6667.
Wrong: 4 out of 6 is a share, not a probability.
Let's look back.
First: binomial: yes or no.
Then: write the rule.
Then: count the orders.
Then: one order's chance.
Then: multiply.
Then: check.
Then: pick the choice.
Last: why not the others.