What to expect
How many heads to expect in 100 flips, and the expected value of a die: the average you get over many tries.
Probability tells you how likely something is on one try. But what happens over many tries? If you flip a coin 100 times, you will not get exactly 50 heads every time, but you will get close to 50. During the Second World War, a mathematician named John Kerrich was held in a prison camp in Denmark, and he used the time to flip a coin 10,000 times. He got 5,067 heads, very close to half. This idea, what to expect over many tries, is used by insurance companies, by casinos, and by anyone who wants to know if a game is fair.
How many times to expect something
If something has a probability of 1/6, then over many tries it happens about one sixth of the time. So to find how many times you expect it, multiply the probability by the number of tries.
Remember
Expected number of times = probability × number of tries
Example 1
You roll a die 120 times. About how many times do you expect to get a 3?
- The probability of a 3 is 1/6.
- 1/6 of 120 is 120 ÷ 6 = 20.
Answer: About 20 times
To find a fraction of a number, divide by the bottom and multiply by the top. For 3/4 of 80: 80 ÷ 4 = 20, and 20 × 3 = 60.
Example 2
This spinner has 5 equal parts: 2 green, 2 blue and 1 yellow. You spin it 200 times. About how many times do you expect green?
- Green is 2 of the 5 parts: 2/5.
- 200 ÷ 5 = 40, and 40 × 2 = 80.
Answer: About 80 times
Example 3
A bag has 3 red and 1 blue marble. You take a marble, look at it and put it back. You do this 60 times. About how many times do you expect red?
- Red is 3 of the 4 marbles: 3/4.
- 60 ÷ 4 = 15, and 15 × 3 = 45.
Answer: About 45 times
The expected value: the average result
Now think about the number a die shows. Over many rolls, what is the average? Each face comes up equally often, so the average is the average of the six faces. This average is called the expected value.
Example 4
What is the expected value of one roll of a die?
- Add the faces: 1 + 2 + 3 + 4 + 5 + 6 = 21.
- Divide by the number of faces: 21 ÷ 6 = 3.5.
Answer: 3.5
You can never roll a 3.5. That is fine. The expected value is not a result you will see. It is the average of many results. Roll a die 1,000 times, add all the numbers and divide by 1,000, and you will get something very close to 3.5.
Prizes and probabilities
When the results are not equally likely, multiply each result by its probability, and add.
Remember
Expected value = (each result × its probability), all added together
Example 5
In a game you win 20 points with probability 1/4. Otherwise you win 0 points. What is the expected number of points?
- 20 points times 1/4: 20 × 1/4 = 5.
- 0 points times 3/4: 0.
- Add: 5 + 0 = 5.
Answer: 5 points
This also tells you if a game is fair. A ticket costs $2. With probability 1/10 it wins $10, otherwise nothing. The expected win is 10 × 1/10 = 1, so $1. You pay $2 to win $1 on average, so each ticket loses $1 on average. Lotteries and casino games are built like this: on average the player loses, and that is how they make money.
Adding expected values
Expected values can be added. If you roll two dice and add the numbers, each die gives 3.5 on average, so the total gives 3.5 + 3.5 = 7 on average. This matches the table of all 36 results of two dice (in the lesson on counting outcomes), where 7 is the middle and most common total.
Example 6
You flip a coin 5 times. What is the expected number of heads?
- One flip gives 1 head with probability 1/2, so 0.5 heads on average.
- 5 flips: 5 × 0.5 = 2.5.
Answer: 2.5 heads
Try it yourself
1.You flip a coin 300 times. About how many tails do you expect?
Show the answer
About 150
Tails has probability 1/2, and half of 300 is 150.
2.A spinner has 8 equal parts, and 3 of them are red. You spin it 40 times. About how many times do you expect red?
Show the answer
About 15
3/8 of 40: 40 ÷ 8 = 5, and 5 × 3 = 15.
3.A spinner has 4 equal parts, numbered 2, 4, 6 and 8. What is the expected value of one spin?
Show the answer
5
2 + 4 + 6 + 8 = 20, and 20 ÷ 4 = 5.
4.You roll three dice and add the numbers. What is the expected value of the total?
Show the answer
10.5
Each die gives 3.5 on average: 3.5 + 3.5 + 3.5 = 10.5.
5.In a game you win 12 points with probability 1/3, and 0 points otherwise. What is the expected number of points?
Show the answer
4
12 × 1/3 = 4, and the 0 adds nothing.
6.A ticket costs $1. With probability 1/100 it wins $50, otherwise nothing. On average, how much do you win or lose per ticket?
Show the answer
You lose $0.50
The expected win is 50 × 1/100 = 0.5, so $0.50. You paid $1, so you lose $0.50 on average.
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Play Probability: 20 seconds per question, and it gets harder as you go.