Independent Events

Lesson 6.2.2

Starter 6.2.2• State in writing whether each of these pairs of

events are disjoint. Justify your answer. If the events are disjoint, calculate P(A or B)– A single fair die is rolled.

• Event A is that an even number comes up. • Event B is that an odd number comes up.

– Two fair coins are tossed.• Event A is that the first coin comes up heads.• Event B is that the second coin comes up heads.

– One card is drawn from a deck of 52 playing cards. • Event A is that the card is a heart. • Event B is that the card is a face card.

Objectives

• Describe what it means to say that two events are independent.

• Use the multiplication rule for independent events to answer “and” probability questions.

• Answer “at least” probability questions through the use of complementary probability.

Independent Events

• Two (or more) events are considered to be independent if the fact that one event occurs does not change the probability that the other will occur.

• If two events A and B are independent, then:P(A and B) = P(A) x P(B)

– This is called the “independent and” rule

• For example, the probability of flipping two coins and getting two heads is:

P(heads and heads) = (.5)(.5) = .25

Example• A bag contains 3 red marbles and 7 black

marbles. One marble is drawn and its color noted. It is then put back in the bag. Another marble is drawn and its color noted.– Event A is: the first marble is red– Event B is: the second marble is red

• Are the two events independent?• Yes:

– P(A) = 0.3. – Because the marble was replaced, P(B) is also 0.3.– The success or failure of event A had no effect on the

probability of success of event B– So P(A and B) = P(both red) = (.3)(.3) = .09

Example• A bag contains 3 red marbles and 7 black

marbles. One marble is drawn and its color noted. It is then set aside. Another marble is drawn and its color noted.– Event A is: the first marble is red– Event B is: the second marble is red

• Are the two events independent?• No: P(A) = 0.3 but P(B) is no longer 0.3

because the marble was not replaced.– The success or failure of event A changes the

probability of success of event B– So we cannot compute P(A and B) by just multiplying.

• We will do “conditional and” computations later.

Calculating “at least” Probabilities• If 3 coins (a quarter, a nickel and a dime)

are tossed, what is the probability that at least one coin comes up heads?– Event A: the quarter is heads– Event B: the nickel is heads– Event C: the dime is heads

• It seems we are interested in P(A or B or C), so we just add:

P(A) + P(B) + P(C) = .5 + .5 + .5 = 1.5 !!!• What’s wrong?

– The events are not disjoint, so the addition rule does not apply

Answer the question empirically• Use the calculator’s Probability Simulator

to toss a coin 3 times, hoping to get AT LEAST one head.

• Do 10 more trials, keeping track of successes and failures.– Report to me the number of successes out of

11 trials.– I will combine class results to get our

empirical probability.

• Next, let’s approach the problem theoretically:

Using Complements to Answer “at least” Questions

• Stand the question on its head:– What is the probability of getting no heads?

• So we are asking the complementary question

– But that means we got all three coins to come up tails, so what is the probability of 3 tails?

– Now we can use the “and” rule because the coins are independent.

• P(tails and tails and tails) = .5 x .5 x .5 = .125

– Then the probability we want is the complement of the probability we just found, so

• P(at least one head) = 1 – .125 = .875

Another Example• Three coins are tossed. Calculate the

probability of getting exactly two heads by application of the “and” and “or” rules.– Hint: Write the sample space to see how many ways

there are to win this bet. Then do the math.– P(H and H and T) = (.5)(.5)(.5) = .125– P(H and T and H) = (.5)(.5)(.5) = .125– P(T and H and H) = (.5)(.5)(.5) = .125– These are disjoint events, so add:

.125 + .125 + .125 = .375

– Note that this is the decimal equivalent of 3/8 and that there are 3 out of 8 branches on your diagram that have exactly two heads

Objectives

• Describe what it means to say that two events are independent

• Use the multiplication rule for independent events to answer “and” probability questions

• A “at least” probability questions through the use of complementary probability

Homework

• Read pages 331 – 335

• Do problems 24 – 27