Multiplication Rule in Probability

In summary, the conversation discusses two probability questions, one involving selecting birthdays and the other involving dice rolls. The first question has an incorrect answer in the textbook, which can be corrected by using the logic of adding the probabilities of different arrangements. The second question has a correct answer, which can be obtained by considering each die roll separately.
  • #1
odolwa99
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Homework Statement



In order to highlight the problem I'm having here I have posted two questions, the answer for the second matches the textbook answer, the first does not. The wording of both questions appears to be the same. Have I gone wrong in the 1st question, or is the book incorrect?

Many thanks.

Q. 1. Three people were selected at random and asked on which day of the week their next birthday was falling. What is the probability that only one of the birthdays falls on a Sunday.

Q. 2. A fair die is thrown 3 times. Find the probability that there will be exactly one 6.

Homework Equations



The Attempt at a Solution



Attempt 1: [itex]P(X)=\binom{3}{1}\binom{1}{7}^1\binom{6}{7}^2= \frac{108}{343}[/itex]

Ans. 1.: (From textbook): [itex]\frac{36}{343}[/itex]

Attempt 2: [itex]P(X)=\binom{3}{1}\binom{1}{6}^1\binom{5}{6}^2= \frac{25}{72}[/itex]
 
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  • #2
odolwa99 said:

Homework Statement



In order to highlight the problem I'm having here I have posted two questions, the answer for the second matches the textbook answer, the first does not. The wording of both questions appears to be the same. Have I gone wrong in the 1st question, or is the book incorrect?

Many thanks.

Q. 1. Three people were selected at random and asked on which day of the week their next birthday was falling. What is the probability that only one of the birthdays falls on a Sunday.

Q. 2. A fair die is thrown 3 times. Find the probability that there will be exactly one 6.

Homework Equations



The Attempt at a Solution



Attempt 1: [itex]P(X)=\binom{3}{1}\binom{1}{7}^1\binom{6}{7}^2= \frac{108}{343}[/itex]

Ans. 1.: (From textbook): [itex]\frac{36}{343}[/itex]

Attempt 2: [itex]P(X)=\binom{3}{1}\binom{1}{6}^1\binom{5}{6}^2= \frac{25}{72}[/itex]

What is the *logic* you used in getting your answer? In other words, why do you write what you did write? (BTW: I get your answer.)
 
  • #3
In Q.1 The arrangement is P(S, F, F) + P(F, S, F) + P(F, F, S). Where F is fail, i.e. not the selected day, so 6/7. And S is success, i.e. the selected day, so 1/7. Then multiply the success as shown in the 1st sentence and add the 3 totals for the answer.

The same logic applies with question 2, except now the odds are that F = 5/6 & S = 1/6.

For the books answer to be correct, factor only a successful day for one outcome, and ignore the remaining 2. I'm assuming that this is what the question is aiming for?

In the second question, 3 separate die rolls means that 3 separate probabilities will be accounted for, not just the 1st.
 
  • #4
You answers are correct. (Btw, it would be better not to use the same notation for both combinatorials and fractions. For the fractions use \frac{}{}.)
 
  • #5
Ok, so the book is definitely wrong? Thanks.
 

Related to Multiplication Rule in Probability

1. What is the Multiplication Rule in Probability?

The Multiplication Rule in Probability is a mathematical rule used to calculate the probability of two or more independent events occurring together. It states that the probability of two or more independent events occurring together is equal to the product of their individual probabilities.

2. How is the Multiplication Rule used in Probability?

The Multiplication Rule is used by multiplying the probabilities of each individual event together. For example, if the probability of event A is 0.5 and the probability of event B is 0.3, the probability of both events occurring together would be 0.5 x 0.3 = 0.15.

3. What is the difference between independent and dependent events in the context of the Multiplication Rule?

Independent events are events that do not affect each other's outcomes. In the context of the Multiplication Rule, this means that the probability of one event occurring does not affect the probability of another event occurring. Dependent events, on the other hand, are events where the outcome of one event can affect the outcome of another event. In this case, the Multiplication Rule would not apply.

4. Can the Multiplication Rule be applied to more than two events?

Yes, the Multiplication Rule can be applied to any number of independent events. The process remains the same - simply multiply the probabilities of each event together to find the probability of all events occurring together.

5. How is the Multiplication Rule related to the Addition Rule in Probability?

The Multiplication Rule and the Addition Rule are two fundamental principles of probability that are often used together. While the Multiplication Rule is used to calculate the probability of independent events occurring together, the Addition Rule is used to calculate the probability of either one or another event occurring. They are complementary rules that help to calculate probabilities in different scenarios.

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