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B. Probability of Multiple Events
 
 


For questions involving single events, the formula for simple probability is sufficient. For questions involving multiple events, the answer combines the probabilities for each event in ways that may seem counter-intuitive. The following strategy is excellent for acquiring a better feel for probability questions involving multiple events or for making a quick guess if time is short. We will focus on questions involving two events.

  • If two events have to occur together, generally an "and" is used. Take a look at Statement 1: "I will only be happy today if I get email and win the lottery." The "and" means that both events are expected to happen together.

  • If both events do not necessarily have to occur together, an "or" may be used as in Statement 2, "I will be happy today if I win the lottery or have email."

Consider Statement 1. Your chances of getting email may be relatively high compared to your chances of winning the lottery, but if you expect both to happen, your chances of being happy are slim. Like placing all your bets at a race on one horse, you've decreased your options, and therefore you've decreased your chances. The odds are better if you have more options, say if you choose horse 1 or horse 2 or horse 3 to win. In Statement 2, we have more options; in order to be happy we can either win the lottery or get email.

The issue here is that if a question states that event A and event B must occur, you should expect that the probability is smaller than the individual probabilities of either A or B. If the question states that event A or event B must occur, you should expect that the probability is greater than the individual probabilities of either A or B. This is an excellent strategy for eliminating certain answer choices.

These two types of probability are formulated as follows:

Probability of A and B
P(A and B) = P(A)
× P(B).

In other words, the probability of A and B both occurring is the product of the probability of A and the probability of B.

Probability of A or B
P(A or B) = P(A) + P(B).

In other words, the probability of A or B occurring is the sum of the probability of A and the probability of B (this assumes A + B cannot both occur). If there is a probability of A and/or B occurring, then you must subtract the overlap.


Look at the following examples.

Example 4

If a coin is tossed twice, what is the probability that on the first toss the coin lands heads and on the second toss the coin lands tails?
a) 1/6
b) 1/3
c) ¼
d) ½
e) 1

Solution
First note the "and" in between event A (heads) and event B (tails). That means we expect both events to occur together, and that means fewer options, a less likely occurrence, and a lower probability. Expect the answer to be less than the individual probabilities of either event A or event B, so less than ½. Therefore, eliminate d and e. Next we follow the rule P(A and B) = P(A)
× P(B). If event A and event B have to happen together, we multiply individual probabilities. ½ × ½ = ¼. Answer c is correct.

NOTE: Multiplying probabilities that are less than 1 (or fractions) always gives an answer that is smaller than the probabilities themselves.

Example 5

If a coin is tossed twice what is the probability that it will land either heads both times or tails both times?
a)1/8
b)1/6
c)1/4
d)1/2
e)1

Solution
Note the "or" in between event A (heads both times) and event B (tails both times). That means more options, more choices, and a higher probability than either event A or event B individually. To figure out the probability for event A or B, consider all the possible outcomes of tossing a coin twice: heads, heads; tails, tails; heads, tails; tails, heads. Since only one coin is being tossed, the order of heads and tails matters. Heads, tails and tails, heads are sequentially different and therefore distinguishable and countable events. We can see that the probability for event A is ¼ and that the probability for event B is ¼. We expect a greater probability given more options, and therefore we can eliminate choices a, b and c, since these are all less than or equal to ¼. Now we use the rule to get the exact answer. P(A or B) = P(A) + P(B). If either event 1 or event 2 can occur, the individual probabilities are added: ¼ + ¼ = 2/4 = ½. Answer d is correct.

NOTE: We could have used simple probability to answer this question. The total number of outcomes is 4: heads, heads; tails, tails; heads, tails; tails; heads, while the desired outcomes are 2. The probability is therefore 2/4 = ½.

The following chart summarizes the "and's" and "or's" of probability:

 Probability   Formula   Expectation
 P(A and B)  P(A) × P(B)  Lower than P(A) or P(B)
 P(A or B)  P(A) + P(B)  Higher than P(A) or P(B)

Additional Example??


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C. Independent and Dependent Events


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