math probability word problem

Question

11. Mr. Wu’s biology class is writing group reports about

the Hawaiian Islands. He has a total of 10 groups.

Mr. Wu will assign an island and a topic at random

to each of the groups. No two group assignments will

be the same.

Five Islands: Hawaii, Maui, Lanai, Oahu,

or Kauai

Topics: Plant Life or

Animal Life

Linda and Kim are in different groups, but they both

want to write about Kauai. What is the probability

that both Linda’s and Kim’s groups will receive Kauai

assignments?

A. 1/5

B.1/45

C. 1/50

D. 1/90

can someone explain HOW to do this problem, not just say that B is the answer?

Answer 1

This problem is an exercise in combining probabilities. Specifically, dealing with independent events vs dependent events. Two events are independent if the outcome of one event does not influence the probability of the other event. For example, flipping a coin. If we flip a coin and get heads on the first toss, the probability that we get heads again on the second toss is still 50%. The fact that the first toss was heads does not impact the results of the second toss.

Two events are dependent if the outcome of one event does influence the probability of the other event. For example, pulling different color marbles out of a bag without replacement. If we have 5 red and 5 green marbles, the probability of pulling red on the first draw is trivial (it is simply 5/10 or 1/2). What is the probability that the second marble is red? Well, now this depends on what our first draw actually was. If we drew red the first time, there are now 4 red marbles and 5 green marbles remaining, so the probability in that case is 4/9. However, if we drew green the first time, the probability of red on the second draw is 5/9. See how the probability of pulling red on the second draw depends on what was drawn the first time?

Now let's apply that concept to this problem. Since no two group assignments will be the same, this is analogous to pulling marbles without replacement. Let's list the possible group assignments, noting that each may only be used once:

1. Hawaii, plant life
2. Maui, plant life
3. Lanai, plant life
4. Oahu, plant life
5. Kauai, plant life
6. Hawaii, animal life
7. Maui, animal life
8. Lanai, animal life
9. Oahu, animal life
10. Kauai, animal life

The probability Linda's group receives an assignment with Kauai is 2/10 (or 1/5). This is the trivial part of the problem. Now, given that Linda's group received Kauai, what is the probability Kim's group will also receive Kauai? Well now, there are only 9 remaining assignments and 1 out of the 2 Kauai options is gone. So that means there is 1 remaining Kauai out of 9 total assignments, or a 1/9 probability.

Finally, the probability that both of these events occur can be found by multiplying the individual probabilities together.

(1/5)*(1/9) = 1/45

Answer 2

This one can take awhile to interpret all the parts correctly. One important thing to note is that the teacher is assigning a different assignment combination to each of 10 different groups, and there are exactly 10 possible assignments (5 Hawaiian islands and 2 biological categories).

There are 10 possible assignments and 2 possible ways to be assigned Kauai (Kauai animal life or Kauai plant life).

Consider the probability that one of the girls is assigned Kauai. This is 2/10 = 1/5.

Given that one of the girls is assigned Kauai already, what is the probability that the other girl is also assigned Kauai? If one is already assigned Kauai, then there are only 9 assignments left to choose from, and there is only 1 other Kauai assignment remaining, so the probability is 1/9.

So the probability of both happening together is the product of the two probabilities: 1/5 * 1/9 = 1/45.

Answer 3

Think of the 10 topics (5 islands*2 places= 10) as 10 blank places on a straight line. Now, think of the 10 groups as 10 people trying to form in that line (1 person cannot be in 2 spots since no 2 group assignments will be the same).

The number of ways for 10 people forming a line is 10!. This is the denominator for the calculation of probability.

Now Linda and Kim (from 2 different groups) both want to write about Kauai. Note that Kauai has 2 topics. so there are 2! or 2 ways of this being assigned (i.e. Linda-- could either do Kauai-plant or Kauai-Animal and Kim will take what Linda doesn't). Now the other 8 groups could have any of the other 8 topics--- this is like letting 8 people fall in line, so there are 8! ways. By multiplication rule, the number of possible ways that that both Linda’s and Kim’s groups will receive Kauai assignments is 2!8!. This is the numerator for the calculation of probability.

Finally, to solve for the probability, we just take the ratio = (2!8!)/(10!) = 1/45

Answer 4

This is a balls in boxes problem. And I made an error in my previous answer.

Mr. WU has 10 "balls", assignments...5 islands and 2 topics.

He will distribute those "balls" in 10 boxes (student groups.

Since no two groups will get the same assignment, he has 10! ways to distribute the "balls",

10 ways for the first group, 9 ways for the 2nd, etc.

Of those ways only 2*1*8! will have 2 particular groups with the same island...2 for the first group, 1 for the 2nd and then 8! for the remaining groups.

Probability = 2*8!/10!=2/(10*9)=1/45

Answer 5

This is a balls in boxes problem.

The teacher has 10 different "balls", i.e. 10 different assignments (5 islands * 2 life forms).and 10 different "boxes" (groups).

He distributes those assignments into 10 different groups. There are 10¹⁰ ways to do that distribution...10 ways to the first box, 10 ways to the 2nd box and so on.

But in order for 2 groups to get the same assignment he has only 2 ways he can fill the first box, only one way to fill the 2nd box and 10⁸ ways to fill the remaining boxes.

Probability = 2*10⁸ / 10¹⁰ = 1/50

Answer 6

This is an example of sampling without replacement with an extra wrinkle. There are 5x2 = 10 possible combinations. When nothing has been drawn yet, each combination has a 1/10 chance of being selected. However, the extra wrinkle is that both Kauai - Animal Life and Kauai - Plant Life satisfy Linda and Kim's desire for Kauai, so the odds of getting Kauai on the first draw are 1/10 + 1/10 = 2/10 = 1/5.

For the second draw, however, only 9 combinations remain, and only one will be for Kauai, given that one of the two Kauai combinations was picked on the first draw. Thus, the probability of again getting Kauai on the second draw is 1/9, and the probability of drawing twice and getting both Kauai combinations is 1/5 x 1/9 = 1/45.

Answer 7

Magnolia, first we have to understand exactly what is meant by “at random”—is it performed with or without replacement? The answer is probably given by the further statement that no two groups receive the same combination—this means it was performed without replacement. In other words, if you’ll write down all ten combinations of (island, topic) each combination will be dealt to a different group.

So, what is the probability that Kim gets one of the two Kauai combinations?

There are two Kaua’i combinations out of 10 combinations total, so 2/10.

Next, if Kim’s group got one of the two Kaua’i combinations, what is the probability that Linda’s group got the other? There are 9 cases remaining, and only one of them is the other Kaua’i combination. So the probability is (2/10)*(1/9) = 2/90 = 1/45