Worked examples
The question
Question: A coin is flipped 3 times.
What is the probability of exactly 1 head?
1 Binomial: yes or no
3 flips.
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 flips: same chance, each on its own
n is the number of flips: 3.
k is how many heads we want: 1.
p is the chance of a success each time: 1/2.
q is the chance of not: 1 minus p, 1/2.
n = 3 flips, k = 1 heads
p = 12 (head)
q = 1 − p = 12
2 Write the rule
The rule: n C k, times p to the power k, times q to the power n minus k.
P(1 head) = nCk × pk × qn − k
We put the numbers in.
P(1 head) = 3C1 × (12)1 × (12)2
3 Count the orders
3 C 1 counts the orders: the one head can be any of the 3 flips.
That is 3.
3C1 = 31 = 3
Here they are: H is a head, T is a tail.
3 orders.
HTT THT TTH
4 One order's chance
One order, like HTT.
Along one path we multiply: 1/2 for each head, and 1/2 for each tail.
one order = 12 × 14
(12)2 = 14
Top times top, bottom times bottom: 1/8.
one order = 18
5 Multiply
Every order has this same chance.
There are 3 orders, so we multiply by 3.
P(1 head) = 3 × 18
Careful!
That is only one order.
Heads can come in 3 different orders.
Careful: not 18 ✗
3 times 1 is 3.
So it is 3/8.
P(1 head) = 38
6 Check
Let's check another order, TTH.
The same number of each, just in other places.
Multiplied, it is 1/8 again.
TTH: 12 × 12 × 12 = 18
And count it: 3 flips make 8 outcomes, all just as likely.
3 of them have exactly 1 head.
That is 3/8.
So the answer is right!
3 of 2 × 2 × 2 = 8 outcomes: 38
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.
▶ Watch it solved
The question
Question: A die is rolled 4 times.
What is the probability of exactly 2 sixes?
1 Binomial: yes or no
4 rolls.
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 rolls: same chance, each on its own
n is the number of rolls: 4.
k is how many sixes we want: 2.
p is the chance of a success each time: 1/6.
q is the chance of not: 1 minus p, 5/6.
n = 4 rolls, k = 2 sixes
p = 16 (six)
q = 1 − p = 56
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 sixes) = nCk × pk × qn − k
We put the numbers in.
P(2 sixes) = 4C2 × (16)2 × (56)2
3 Count the orders
4 C 2 counts the orders: which 2 of the 4 rolls are the sixes.
4 times 3, over 2 times 1, is 6.
4C2 = 4 × 32 × 1 = 6
Here they are: 6 is a six, N is a roll that is not a 6.
6 orders.
66NN 6N6N 6NN6
N66N N6N6 NN66
4 One order's chance
One order, like 66NN.
Along one path we multiply: 1/6 for each six, and 5/6 for each roll that is not a 6.
one order = 136 × 2536
(16)2 = 136
(56)2 = 2536
Top times top, bottom times bottom: 25/1296.
one order = 251296
5 Multiply
Every order has this same chance.
There are 6 orders, so we multiply by 6.
P(2 sixes) = 6 × 251296
Careful!
That is only one order.
Sixes can come in 6 different orders.
Careful: not 251296 ✗
6 times 25 is 150.
So it is 150/1296.
P(2 sixes) = 1501296
Both 150 and 1296 can be divided by 6.
That makes it simpler.
P(2 sixes) = 150 ÷ 61296 ÷ 6
So it is 25/216.
P(2 sixes) = 25216
As a decimal, to 4 places, that is about 0.1157.
P(2 sixes) ≈ 0.1157
6 Check
Let's check another order, NN66.
The same number of each, just in other places.
Multiplied, it is 25/1296 again.
So the answer is right!
NN66: 56 × 56 × 16 × 16 = 251296
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 P(2 sixes) ≈ 0.1157
▶ Watch it solved
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) = nCk × pk × qn − 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 P(3 successes) = 0.512
▶ Watch it solved
