1. Bernoulli Trials and The Binomial Distribution

2. Definition
Bernoulli Trials are questions which involve the use of success and failure.
Essentially, a Bernoulli trial is an experiment where there are only 2 outcomes ‘success’ and ‘failure’

3. Where do we see Bernoulli trials?
tossing a coin
looking for defective products rolling off an assembly line
shooting free throws in a basketball game or in a competition.
We can see with all these there are only 2 outcomes. Success and Failure.

4. The Binomial Distribution
The Binomial Distribution is a type of Bernoulli trial and when we are using the binomial distribution we use the following formula which is on Pg 33 of the log tables.

5. The following explains the terms used in this formula:
p =the probability of success (as a decimal).
q= the probability of failure (as a decimal)
p + q =1. Therefore, q = 1 – p
n = the number of trials
r = the number of successes we want.

6. Very Important
There are 4 essential features of binomial distributions within Bernoulli trials which must be learned:
1. There must be a fixed number of trials, n
2. The trials must be independent of each other
3. Each trial has exactly 2 outcomes called success or failure
4. The probability of success, p, is constant in each trial

7. Steps in calculating the Binomial distribution:
Write down n (the number of trials)
Calculate p, the probability of success
Calculate q, the probability of failure
Let r = the number of successes we require.
Sub into the formula

8. Question:
A coin is tossed six times, what is the probability of getting exactly four heads?

9. In this question the probability of success and failure are the same.
n= 6, p=0.5, q= r= 4

10. Question: A fair die is thrown 5 times
Question: A fair die is thrown 5 times. Note: as the die is fair the probability of success is: Find the probability of obtaining:
1 Six
3 Sixes
No Six
At least 3 sixes

11. A company is doing a test on the number of smoke alarms which are faulty. It is said that 10% of smoke alarms are faulty. In a box containing 12 smoke alarms. Find the probability that:
None are faulty
Exactly One is faulty
At least 3 are faulty

12. 20% of the items produced by a machine are defective
20% of the items produced by a machine are defective. Four items are chosen at random.
Find the probability that none of the chosen items are defective.
Find the probability that 3 of the chosen items are defective.

13. Question
Lionel Messi scored 40% of his shots during the season. He has a game tonight vs Real Madrid.
Find the probability of success
Find the probability of a failure
Find the probability that he scores for the first time in tonight’s game after 4 shots.
Find the probability that he misses for the first time in tonight’s game after 3 shots.

14. A basketball player has made 80%, of his foul shots during the season
A basketball player has made 80%, of his foul shots during the season. Assuming the shots are independent, find the probability that in tonight's game he:
misses for the first time on his fifth foul shot
makes his first basket on his fourth foul shot

16. Questions which involve the probability of the kth success on the nth Trial.
For example:
What is the probability that he is successful for the second time on the 5th trial.
What is the probability that the basketball player scores his 2nd basket on his 5th shot.

17. Important
What is the probability that the basketball player scores his 2nd basket on his 5th shot.
With these questions it is essential to realise that there are 2 successes.
The First success occurs in the first 4 shots.
2nd Success occurs on his 5th shot.

18. A basketball player scores 60% of the free throws she attempts
A basketball player scores 60% of the free throws she attempts. During a particular game she gets 6 free throws.
What assumptions must be made in order for this to be a Bernoulli trial and use the binomial distribution formula?
What is the probability that she scores exactly 4 of the 6 free throws?

19. What is the probability that she scores for the 2nd time on the 5th free throw?

20. Ronald is his schools best basketball shooter
Ronald is his schools best basketball shooter. He is a 70% free throw shooter. Therefore the probability of him scoring on a free throw is 0.7. What is the probability that Ronald scores his third free throw on his fifth shot?

22. A fair die is thrown repeatedly. The probability of a getting a 6 is 0
A fair die is thrown repeatedly. The probability of a getting a 6 is What is the probability that the 3rd six will occur on the 8th throw.