3 Math Questions - Probability

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1

<p>1.) A bag contains 8 white marbles, 8 blue marbles, 7 red marbles, and 6 yellow marbles. What is the least number of marbles that can be drawn from the bag, so that 3 of the same color marbles will be drawn?</p>

<p>(A) 6
(B) 9
(C) 12
(D) 13
(E) 15</p>

<p>Answer explanation in book: B
W B R Y
W B R Y
(W)
The ninth marbles gives the 3 of the same color
My Question: They asked for the "least" number of marbles that can be drawn...so shouldn't the answer be 3? (even though this answer is NOT in the options available). Example: I could pick 3 whites. Done. Please explain why my logic is wrong.</p>

<p>2.) If a fair die is thrown three times, what is the probability that a 5 comes up exactly two times?</p>

<p>Answer: 5/72
There are three possibly ways as above. Therefore, P = 3 (1/6) (1/6) (5/6) = 5/72
My Question: Please explain. I do not understand how to solve this or understand the book explanation. Please explain in-depth why we have to multiple "3" and "5/6". </p>

<p>3.) If each of 8 boys played a game of chess with each of 6 girls, and then each girl played a game with each of the other girls, which of the following could be the total number of games played?</p>

<p>(A) 63
(B) 65
(C) 69
(D) 75
(E) 78</p>

<p>Answer: A
There are 48 games between 8 boys and 6 girls, and 15 games between the girls. 8 x 6 = 48 and 5 + 4 + 3 + 2 + 1 = 15 games. Therefore 48 + 15 = 63 games.
My Question: I have two issues with this problem. First, isn't this a combination problem? For the first game (boys-girls), shouldn't we divide 48 by 2? Second issue, the girl-girl game is a combination problem (6x5 divided by 2 = 15), so how come the boy-girl game isn't a combination problem? A boy-girl game is the same as a girl-boy game.</p>

2

<p>Hi Morton, I have to say I am not very good at this also, but I’ll still try if I can help.</p>

<p>Question 1: you are asking why the answer should not be 3. Well, if you are lucky enough, 3 times drawing the marbles from the bag CAN do. However, the possibility of that to happen is very low, say, less than 100%. I guess the question is asking for the number of marbles that are needed to be drawn out that GUARANTEES 3 of the same colour. Thus, the answer is 9 because, say you are very unlucky, for the first 4 times you pick the marbles, you get 4 of the DIFFERENT colours, and the next 4 times you pick, again, 4 DIFFERENT colours. you have already picked 8 times and the 9th should be one of the 4 colours that makes up a combination of 3 same colour marbles. That’s the written explanation that I guess you understand. But the key of this question is how many can GUARANTEE a combination of 3 similar colour marbles. </p>

<p>Question 2: I know the reason why they multiple by 3 and 5/6, but I don’t really understand why it is necessary to multiple by 3. Anyway, I am going to write it down here. There are 3 ways, as the answer says, #1: you get 5 the 1st and 2nd time, #2: you get 5 the 2nd and 3rd time, #3: you get 5 the 1st and 3rd time. That is why they multiple it by 3. The 5/6 is easier. The probability that you get 2 times the 5 is (1/6)(1/6), since you roll 3 times and the remaining time MUST NOT be 5, they multiple 5/6 in, which is the probability that the 5 DOES NOT come up. So we have 3(1/6)(1/6)(5/6). There is a general formula for this. </p>

<p>Question 3: the boy-girl is not the same as the girl-girl. Well, I shall not talk about it in terms of combination (because I have not really been taught combination, I learn it by myself so I’m not confident). “Each of the 8 boys plays a game with each of the 6 girls”. It means each boy plays 6 games. we have 8 boys, so the number of games the boys play are 8x6=48. There is not a need to consider whether girl boy or boy girl count as 1 or 2. It is different from the girl-girl because the girls plays with EACH OTHER. So now if we take each one of the girls plays with 5 others, the repetition of girl A vs B and B vs A is included and thus, we need to divide by 2. </p>

<p>I know I am not very good at explaining, but I hope it help. If you have any further questions, please feel free to ask. Thank you for posting the question by the way. Good luck and have fun!</p>

3

<p>I agree with waytosucess’s interpretation of question 1, and with his explanation for all three problems</p>

<p>As posed, question 1 is ambiguous. Can you check to see if the wording was something like this:</p>

<p>1.) A bag contains 8 white marbles, 8 blue marbles, 7 red marbles, and 6 yellow marbles. What is the least number of marbles that you must draw from the bag to be certain that 3 of the same color marbles are always drawn?</p>

4

<p>@waytosuccess thanks for explaining! I understood your explanations for question 1 and 2.</p>

<p>Can someone please explain the combination problem in question 3?</p>

5

<p>For #3, it sounds like you are ok with the second part. So let me restate the first part. It’s not a combination problem, but rather a simple application of the counting principle.</p>

<p>Suppose you had 8 hats and 6 ties. How many hat-tie sets could you make?</p>

<p>You would have 8 choices for the hats, then 6 for the ties: 8 x 6 = 48. </p>

<p>There’s no reason to now divide that answer by 2. It does not matter whether you pick the hat first and then the tie or the tie first and then the hat. There are still 6 x 8 = 48 pairings and your method of counting them did not “overcount” because your options (the 6 and the 8) came from two DIFFERENT sets.</p>

<p>Compare that to the following: You have 8 necklaces of different colors. You decide to wear two of them. How many ways can you do it? Now you have 8 choices followed by 7 choices FROM THE SAME SET. So 8 x 7 =56 is an overcount by a factor of 2 – it counts red and blue as distinct from blue and red. So now you DO have to divide by 2.</p>

<p>Hope that helps…</p>