Answers:
- (a) 39%
- (b) 0.5625
- (c) 0.7692 (approximate)
- (d) Yes there is a connection. The events are dependent.
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Explanation for part (a)
The table says there are 39 students who do not carry a backpack out of 100 total.
39/100 = 0.39 = 39% of the students do not carry a backpack.
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Explanation for part (b)
The phrasing "if a junior is chosen" is the same as saying something like "given we know a junior has been chosen". The word "given" is a key term in conditional probability questions.
We focus entirely on the juniors only. Ignore everyone else.
I recommend either using a highlighter to mark the "junior" column, or using two sheets of paper to cover the other columns up.
We have 18 juniors that carry a backpack out of 32 juniors total.
18/32 = 9/16 = 0.5625 is the probability the junior has a backpack. This value is exact and hasn't been rounded.
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Explanation for part (c)
This is similar to the previous part. The "given" this time is we know 100% the student selected doesn't have a backpack.
Focus solely on the "no backpack" row. There are 14 juniors and 16 seniors in this row. There are 14+16 = 30 juniors or seniors that don't have a backpack. This is out of 39 people who don't have a backpack.
30/39 = 0.7692 which is approximate. Round this value however needed.
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Explanation for part (d)
Define these events
- A = person is a freshman
- B = person has a backpack
Then,
- P(A) = 11/100 = 0.11
- P(B) = 61/100 = 0.61
- P(A and B) = 8/100 = 0.08
If events A and B were independent, then P(A and B) = P(A)*P(B) would be a true equation.
P(A)*P(B) = 0.11*0.61 = 0.0671 exactly without any rounding done to it
This does not match with P(A and B) = 0.08
Therefore, P(A and B) = P(A)*P(B) is false. Events A and B are not independent. They are dependent in some way.
Use this logic to explore the connections between the other grade levels and their status of "backpack" vs "no backpack". You should find that those connections are also dependent.
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Alternative explanation for part (d)
Once again I'll define these two events:
- A = person is a freshman
- B = person has a backpack
If the events were independent, then these two equations must be true
- P(A given B) = P(A)
- P(B given A) = P(B)
The table says P(A) = 0.11 as calculated earlier.
P(A given B) = probability a freshman is chosen, given they have a backpack
P(A given B) = (8 freshmen with backpacks)/(61 people with backpacks)
P(A given B) = 8/61
P(A given B) = 0.1311 approximately
This does not match up with P(A) = 0.11 calculated earlier.
We have shown that P(A given B) = P(A) is false in this case, which must mean the events are dependent somehow. Having prior knowledge of the student having a backpack (or not) changes the probability of P(A).
You should find that P(B given A) = P(B) is false here as well. I'll let you explore this connection, and the other paired connections.
To conclude part (d) in one sentence: Yes there appears to be a connection between grade level and whether the student carries a backpack or not.