114 7.3. Expected Value and Tree Diagrams — Mathematics for Public and Occupational Health Professionals

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Expected Value

An expected gain or loss in a game of chance is called expected value. The concept of expected value is closely related to a weighted average. Consider the following situations.

In the first situation, to find the expected value, we multiplied each payoff by the probability of its occurrence, and then added up the amounts calculated for all possible cases. In the second example, if we consider our test score a payoff, we did the same. This leads us to the following definition.

Expected Value: If an experiment has the following probability distribution,

Payoff x1   x2   x3  ···   xn
Probability p(x1)   p(x2)   p(x3)   ···   p(xn)

 

So on average, there are 1.56 children to a family.

 

 

 

 

Let U be the event that the door has been unlocked and L be the event that the door has not been unlocked. We illustrate with a tree diagram.

 

 

 

Practice questions

1. In a European country, 20% of the families have three children, 40% have two children, 30% have one child, and 10% have no children. On average, how many children are there to a family?

2. A local community center plans to raise money by raffling a $500 gift card. A total of 3000 tickets are sold at $1 each. Find the expected value of the winnings for a person who buys a ticket in the raffle.

3. A $1 lottery ticket offers a grand prize of $10,000; 10 runner-up prizes each paying $1000; 100 third-place prizes each paying $100; and 1,000 fourth-place prizes each paying $10. Find the expected value of entering this contest if 1 million tickets are sold.

4. A game involves drawing a single card from a standard deck of 52 cards. One receives 75 cents for an ace, 25 cents for a king, and 5 cents for a red card that is neither an ace nor a king. If the cost of each draw is 15 cents, what is the expected value of the game?

5. A basketball player has an 80% chance of making a basket on a free throw. If he makes the basket on the first throw, he has a 90% chance of making it on the second. However, if he misses on the first try, there is only a 70% chance he will make it on the second. If he gets two free throws, what is the probability that he will make at least one of them?

6. A die is rolled until a one (1) shows. What is the probability that a one will show in at most four rolls?

7. You forget to set your alarm 60% of the time. If you hear your alarm, you will turn it off and go back to sleep 20% of the time. Even if you do get up on time, you will be late getting ready about 30% of the time. Under these circumstances, what is the probability that you will be late to class in the morning?

8. Your friend wants to take the Ontario Real Estate License exam, which has a pass rate of about 60%. If a person fails the exam, their success rate improves to about 70% on the second try, and 75% on the third try. What is the probability that your friend will pass the exam in at most three tries?

 

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