Three Probability Questions That Are Hard To Solve!

Question

I am having Difficulty answering these three probability questions. If you could, could you help? Thanks!

1. A group of people were asked “What time do you prefer to see a movie? The two way table below represents the results by their age.

 

	16-20	21-25	26-30	Over 30	Totals
Morning	4	2	12	25	43
Afternoon	8	12	18	32	70
Evening	28	34	28	11	101
Late Night	34	18	21	4	77
Totals	74	66	79	72	291

 

What is the approximate probability that a person will be between the ages of 26 -30 given they prefer evening movies?

2.Let Events A and B be described as follows:

 

1. P(A) = doing yard work
2. P(B) = it raining

 The probability that it will rain this weekend is 68% The probability of doing yard work this weekend and it raining is15%. If the probability of doing yard work is 53%, are doing yard work and it raining independent?

A.Yes, because because P(A| B) = 0.28 and the P(B) = 0.68 are not equal.

B.Yes, because P(A | B) = 0.22 and the P(A) = 0.53are not equal.

C.No, because P(A| B) = 0.28 and the P(B) = 0.68 are not equal.

D.No, because P(A | B) = 0.22 and the P(A) = 0.53are not equal.

3.Let Events A and B be described as follows:

 

1. P(A) = ordering room service
2. P(B) = watching television

 The probability of watching television is 42% and so is the probability of ordering room service. The probability of ordering room service and watching television is 17.64%.

 

Which statement is accurate about these two events?

A.The two events are not independent because P(A)P(B) = 0.074 and the P(A and B) = 0.1764 are not equal.

B.The two events are independent because P(A | B) = .42 and the P(B) = 0.42 are equal.

C.The two events are not independent because P(A)+P(B) = 0.5964 and the P(A and B) = 0.1764 are not equal.

D.The two events are independent because P(A | B) = .42 and the P(A) = 0.42 are equal.

Answer 1

Problem 1. We are asked what is the proability that an individual is between 26 and 30 GIVEN they prefer evening movies. So we are only considering those who prefer evening movies, and we are ignoring everything else. Of the 101 people who prefer evening movies, 28 of them are between 26 and 30 years old. P(26-30| evening) = 28/101= .277

Problem 2. Two events are independent if the probability of one event doesn't affect the probability of the other event: if a few raindrops don't affect the proability that we do yardwork. This is NOT the case.Let A be the event we do yardwork, and B be the event it rains: P(A|B) = P (A∩B)/ P(B) = .15/.68 = .22 ≠ .53. The correct answer is D.

Problem 3. This is just like the previous problem. Let A be the event Watch TV, and let B be the event room service. P(A|B) = P(A·∩B) / P(B) = .1764 / .42 = .42 = P(A). The answer is D.

Here is another way to think about independence. Two events are independent if P(A∩B)=P(A)P(B). .42(.42) = .1764. But the multiple choice answers do not fit with this approach.