14.2: Analyzing Probability Problems

You must know the rules if you are to calculate probabilities.

Summary of the Rules for Calculating Probabilities

1. Events that are Certain to Occur: If A is certain to be true, Pr(A) = 1.
2. Events that are Certain not to Occur: If A is certain to be false, Pr(A) = 0.
3. Negations: Pr(~A) = 1- Pr(A).
4. Disjunctions with Incompatible Disjuncts: If A and B are incompatible, Pr(A or B) = Pr(A) + Pr(B).
5. Conjunctions with Independent Conjuncts: If A and B are independent, Pr(A & B) = Pr(A) x Pr(B).
6. Disjunctions: Pr(A or B) = Pr(A) + Pr(B) - Pr(A & B).
7. Definition of Conditional Probability: Pr(A|B) = Pr(A & B)/Pr(B).
8. Conjunctions: Pr(A & B) = Pr(A) x Pr(B|A).

How to Approach a Problem

The key is to analyze a problem before you begin writing things down. The first question to ask yourself is: Am I calculating the probability of a negation, a disjunction, or a conjunction? The answer to this will tell you which rule is relevant to the problem; if you get this right, you are well on your way to a successful solution. In a complicated problem, you may have to use several of these rules in your calculations, but always begin by asking which rule applies first.

Begin by thinking through these steps:

1. If the sentence is a negation, use Rule 3.
   • Find the probability of the sentence that is being denied and subtract it from 1.
2. If it is a disjunction
   • Are the disjuncts incompatible? If so, use Rule 4.
   • Are the disjuncts compatible? If so, use Rule 6.
3. If it is a conjunction
   • Are the conjuncts independent? If so, use Rule 5.
   • Are the conjuncts dependent (= not independent)? If so, use Rule 8.

The tree diagram in Figure 14.2.1 represents the same information pictorially.

Examples of Problem Analysis

Problem A. Suppose that you have a standard deck of 52 cards. You will draw a single card from the deck. What is the probability of drawing either an ace or a jack? Analysis of the problem:

1. You want to know about the probability of drawing an ace or drawing a jack, so you have a disjunction. The first disjunct is, “I get an ace,” and the second disjunct is, “I get a jack.” We could symbolize this as (A or J).
2. Are the disjuncts incompatible? Well, if you draw an ace you cannot also draw a jack (on that same draw). Getting an ace excludes getting a jack (and getting a jack excludes getting an ace). So, the disjuncts are incompatible, and you use R4 (the rule of disjunctions with incompatible disjuncts).

Figure \(\PageIndex{1}\): Tree Diagram of Probability Problem Analysis

3. The rule says to add the probabilities of the two disjuncts: Pr(A or J) = Pr(A) + Pr(J)
4. There are exactly four aces out of 52 cards, so Pr(A) (the probability of drawing an ace) is 4/52 (which reduces to 1/13). There are also four jacks, so Pr(J) is the same as that of drawing an ace, namely 1/13.
5. Rule 4 tells us to add these probabilities: Pr(A or J) = 1/13 +1/13(= 2/13).

Problem B. Suppose that you have a standard deck of 52 cards. You will draw a single card from the deck. What is the probability of drawing either a jack or a heart? Analysis:

1. You want to know about drawing a jack or drawing a heart, so you again have a disjunction. The first disjunct is, “I get a jack” and the second disjunct is, “I get a heart.” We symbolize this as (J or H).
2. Since you have a disjunction, the relevant rule will be one of the two Disjunction Rules. Which one it is depends on whether the disjuncts are incompatible.
3. Are the disjuncts incompatible? Well, if you draw a jack, does that exclude drawing a heart? No. You might draw the jack of hearts. So, the disjuncts are not incompatible, and you must use Rule 6 (the general disjunction rule).
4. This rule says to add the probabilities to the two disjuncts, but then “subtract out the overlap.” In other words, you must subtract out the probability that you get both a jack and a heart, and this is just the probability of getting the jack of hearts. So we have Pr(J or H) = Pr(i) + Pr(H) - Pr(J & H).
5. There are four jacks out of 52 cards, so Pr(J), the probability of drawing a jack is 4/52. And there are 13 hearts, so Pr(H), the probability of drawing a heart is 13/52. Finally, there is just one possibility for getting a jack and a heart, namely the jack of hearts, so Pr(J & H) is 1/52.
6. The General Disjunction Rule then tells us Pr(J or H) = 4/52 + 13/52 - 1/52 (we won’t worry about actually calculating such things until we get the basic concepts down—and even then you can use a calculator).

Exercises

1. The chances of there being two bombs on a plane are very small, so when I fly, I always take along a bomb. —Laurie Anderson. What should we make of Anderson’s advice (given what we have learned thus far)?
2. What is the numerical value of Pr(A|A)? Explain why your answer is correct.
3. Suppose you are going to flip a fair coin. Which of the possible sequences is/are the most likely?
   1. HHHHTTTT
   2. HTHTHTHT
   3. HTHHTHTH
   4. HTHHTHTHT
   5. No one of these is any more likely than the others.
4. Suppose that you are about to turn over four cards from the top of a standard deck. Which of the following series of cards (in the order given) is the most likely?
   1. Ace of hearts, king of diamonds, queen of spades, jack of hearts
   2. Ace of heart, king of hearts, queen of hearts, jack of hearts
   3. Ace of hearts, eight of spades, jack of diamonds, four of clubs
   4. No one of these is any more likely than the others.
5. If two sentences are incompatible, then:
   1. They must also be independent.
   2. The truth of one is completely irrelevant to the truth of the other.
   3. They cannot also be independent.
   4. None of the above.
6. You have an ordinary deck of 52 cards. You will draw a card, lay it on the table, then draw another card. (It is important to use the rules in these calculations.)
   1. What is the probability of two kings?
   2. What is the probability of a queen on the second draw given a king on the first?
   3. What is the probability of a king on the first draw?
   4. What is the probability of the king of spades and the king of hearts (in either order)?
   5. What is the probability of a king and a queen?
   6. What is the probability of the jack of diamonds and a spade (where the order in which you get the two doesn’t matter)?
   7. What is the probability of not drawing a five at all?
7. Suppose that I am planning what to do this coming weekend, and the weather forecast is for 40% chance of rain on Saturday and 40% chance of rain on Sunday (40% chance = .40 probability). What is the probability that it will rain sometime or other during the weekend (assume that it’s raining or not on Saturday won’t make it any more or less likely to rain on Sunday)?
8. What is Pr(S|S)? What about Pr(~S|~S). Explain and defend your answers.
9. You and your friend Wilbur are taking a multiple-choice exam (and you are working independently, and your answers are independent). There is exactly one correct answer to each question, and your task is to select it from five possible answers, ‘a’, ‘b’, ‘c’, ‘d’, and ‘e’. You get to the third question and have no idea what the correct answer is, and the same thing happens with Wilbur. You guess ‘a’, and Wilbur guesses ‘c’.
   1. What is the probability that at least one of you got the correct answer?
   2. What is the probability that neither of you got the correct answer [the answer here is not 4/5 x 4/5].
   3. You also guessed on the fourth problem. What is the probability that you got at least one of your two guesses is right?
10. Most automobile accidents occur close to home. Why do you suppose this is true? How could you explain what is involved using the notion of conditional probabilities?