Analytic Geometry Name______
Independent & Dependent Events Date ______Period ______
In the previous unit on counting, you may remember that when a problem has two or more events, and all of them will take place, then it requires the multiplication principle of counting. The same general idea works for probability.
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Events are independent events when the outcome of one event does not influence the outcome of the second event.
When the outcome of one event affects the outcome of a second event, the events are
dependent events.
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Decide whether the following examples are dependent or independent events.
1. Spinning a wheel two times
2. Drawing two cards from a deck if the first card is replaced after being drawn
3. Drawing two cards from a deck if the first card is not replaced
4. Rolling a die and flipping a coin
5. Selecting the racer who will finish first in a race, then selecting who will finish second
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The multiplication rule for independent events is:
So, if you have two independent events (called A and B), then in order to find the probability that A and B will happen, you simply multiply the probability that A will occur by the probability that B will occur.
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Since these two events are independent events (see Question 1 above), the formula for independent events can be used:
P(A and B) = P(A)*P(B) = P(3 on 1st spin)*P(3 on 2nd spin) =
What is the probability of drawing an Ace on three straight draws if the card is replaced in the deck each time it is drawn?
The multiplication rule for dependent events is:
As you will see, the difference between independent and dependent events is that the desired outcomes and/or the total outcomes change for event B.
------A bag contains four gold tokens, five silver tokens, and eight bronze tokens. If two tokens are taken from the bag and not replaced, what is the probability that both tokens are gold?
Once again, notice that there are two events - the two draws of tokens.
Why are these two events dependent events?
P(A and B) = P(A) * P(B assuming A was successful)
= P(1st gold) * P(2nd gold if 1st was successful)
=
=
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What is the probability of drawing three consecutive Clubs from a standard deck of cards if the cards are not replaced after being drawn?
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Suppose there eight runners in a race. What is the probability of correctly selecting the runners who will finish in first and second place?
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Very Important Idea - You should look to multiply in problems where there is more than one event and all will take place - for example, drawing multiple cards, spinning a wheel more than once, picking more than one person, etc.
Practice Problems Name______
3. Two cards are drawn from a standard deck. What is the probability that the first card is
a 9 and the second card is a 10 if the cards are not replaced after being drawn?
4. At a cross country event at Lakeside High School, there are eight runners from Lakeside,
nine runners from Evans, and twelve runners from Greenbrier. What is the probability
that a runner that is NOT from Lakeside wins the race? Assume all runners have an
equal chance to win.
5. Mr. B. has five favorite NFL teams. If there are 32 teams in the NFL, what is the probability that two of his five favorite teams finish in 1st and 2nd place in wins for the
season? Assume all teams have an equal chance and that there can be no ties.
6. A corrupt king has ten men, three women, and two children in his prison. If he will randomly kill two prisoners, what is the probability that both are women?
Note: the children are NOT considered women.
7. Four cards are drawn from a standard deck. If a card is replaced each time it is drawn,
what is the probability of drawing a Spade each time?
8. There are two big horse races taking place on Saturday. A horse owner named Guy has four horses in the first race, and three different horses in the second race. If both races
have fifteen total horses each, what is the probability that a horse owned by Guy wins both races? Assume all of the horses in both races have an equal chance of winning.
9. Two dice are rolled simultaneously. What is the probability that the sum of the dice is
11 or 12?
10. In the movie The Wedding Planner, Steve says, “I only eat the brown M&M’s because I figure they have less artificial coloring.” So, he eats the brown M&M’s, and he throws any other color on the ground. If there are 58 M&M’s in a bag, and eight are brown, what is the probability of getting two brown ones in a row?
11. What is the probability that the last four digits of a person’s phone number are all greater than 5?
------Answers: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.