P - Binomial Probability Lesson

Binomial Probability 

At the National Baseball Batting Contest, the organizers have set up game booths for the contestants. Richard wants to win a large stuffed animal. The game costs $3 to play (each set of 5 fastballs).

The rules of the game are as follows:

• You are pitched 5 fastballs, and you must hit them into a fair zone to count.
• If you successfully hit all 5 pitches, you win a large stuffed animal.
• If you successfully hit 3 or 4 pitches, you win a small stuffed animal.
• If you successfully hit 1 or 2 pitches, you win a bat-shaped pencil.
• If you miss all the pitches, you do not win a prize.

What are the possible outcomes that Richard can have on his 5 swings?

Batting skills would alter the probabilities. Assume no skill is required. There are two possible outcomes every time Richard has a fastball thrown at him, he can either hit it or miss it. So, he has a  chance of hitting each fastball and he has a  chance of missing. The chances of hitting each fastball do not change based on the result of a previous hit or miss - they are all independent.

What is the probability that Richard will hit all 5 pitches? 

Because each swing is independent, the probabilities are multiplied.

What is the probability that Richard will hit exactly 4 pitches? There are 5 possible ways that Richard could do this, each way has the same probability, so the total probability is . See the graphic below for an explanation, notice that the probability for a hit and a miss are the same. The math on the first row has been worked out for you - it is the same on all other rows.

 

What is the probability that Richard will hit exactly 3 pitches? Similar to the question above, there are several ways Richard could do this, in fact, there are 10 possible ways this could happen. The chance for each one is still 1/32, so the total probability is . See the graphic below for further explanation. 

What is the probability that Richard will hit exactly 2 pitches? The answer to this one is exactly the same as the probability for 3 hits. Using the graphic for 3 hits, simply reverse the green and red and you will have the probability for 2 hits.

 

What is the probability that Richard will hit exactly 1 pitch? The answer to this one is exactly the same as the probability for 4 hits. Using the graphic for 4 hits, simply reverse the green and red and you will have the possibility for 1 hit.

 

What is the probability that Richard will have no hits? The answer to this one is exactly the same as the probability for 5 hits.

Expected Value

Expected value can be understood as the average payoff in a game or contest when either is played a large number of times. To find the expected value of a game, we multiply the gain or loss for each possible outcome by its probability. Then we sum up the products.

For each of the examples below, assume Richard tried to play the game 10 times.

What is the expected value for Richard to hit 5 balls?

The probability of successfully hitting 5 balls is 1/32, so if Richard tries 10 times you expect him to hit 5 balls for 10/32 tries. Therefore, the expected value is about 0.3 wins, so you do not expect him to successfully hit 5 balls in the 10 tries.

What is the expected value for Richard to hit 4 balls?

The probability of successfully hitting 4 balls is 5/32, so if Richard tries 10 times you expect him to hit 4 balls for 50/32 tries. Therefore, the expected value is 1.6 wins, so you expect him to win about 1 to 2 times if he tries 10 times.

What is the expected value for Richard to hit 3 balls?

The probability of successfully hitting 3 balls is 10/32, so if Richard tries 10 times you expect him to hit 3 balls for 100/32 tries. Therefore, the expected value is 3.1 wins, so you expect him to win about 3 times if he tries 10 times.

What is the expected value for Richard to hit 2 balls?

The probability of successfully hitting 2 balls is 10/32, so if Richard tries 10 times you expect him to hit 3 balls for 100/32 tries. Therefore, the expected value is 3.1 wins, so you expect him to win about 3 times if he tries 10 times.

What is the expected value for Richard to hit 1 ball?

The probability of successfully hitting 1 ball is 5/32, so if Richard tries 10 times you expect him to hit 1 ball for 50/32 tries. Therefore, the expected value is 1.6 wins, so you expect him to win about 1 to 2 times if he tries 10 times.

What is the expected value for Richard to hit 0 balls?

The probability of hitting 0 balls is 1/32, so if Richard tries 10 times you expect him to hit 0 balls for 10/32 tries. Therefore, the expected value is 0.3, so you do not expect this outcome.

Consider 160 Players

Everyone has the same chance as Richard to win. If 160 people each play the game once, how many large stuffed animals will be won?

The probability of Richard successfully hitting 5 balls and winning a large stuffed animal is 1/32, or about 3%. So when this percentage is multiplied by 160 people, only 5 will be expected to win a large stuffed animal.

If 160 people each play the game once, how many small stuffed animals will be won?

To win a small stuffed animal, you have to successfully hit 3 or 4 balls.

The probability of hitting 3 balls is 10/32 and the probability of hitting 4 balls is 5/32

You try this one:

If 160 people each play the game once, how many bat-shaped pencils will be won?

The answer is the same as the one above it. The probability for 2 hits is 10/32 and the probability for 1 hit is 5/32. Therefore the total probability is 15/32 = 46.9%.  And 46.9 % of 160 = 75.

 

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