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The high temperature of a summer day on Fire Island is [#permalink]

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26 Sep 2006, 10:13

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The high temperature of a summer day on Fire Island is always one of three temperatures: 84, 85 or 86 degrees. If there is an equal chance of each temperature, what is the probability that the median high temperature over a three day period will be 85 degrees?

The high temperature of a summer day on Fire Island is always one of three temperatures: 84, 85 or 86 degrees. If there is an equal chance of each temperature, what is the probability that the median high temperature over a three day period will be 85 degrees?

1) 1/27

2) 2/9

3) 1/3

4) 4/9

5) 13/27

When we are asked to find the median of 3 numbers, we arrange them in increasing order and then pick the middle number. The median of 85 will be obtained when the temperatures are one of: {85, 85, 85} {85, 86, 86} {84, 85, 85} {84, 85, 86}

When will the temperatures be {85, 85, 85}? When every day the high is 85. What is the probability of this happening? 1/3 * 1/3 * 1/3 = 1/27

When will the temperatures be {85, 85, 86}? When the high is 85 on two days and 86 on another. What is the probability of this happening? Probability of 85 on first day is 1/3. Of 85 on second day is 1/3 and of 86 on third day is 1/3. Probability of this is 1/3 * 1/3 * 1/3 = 1/27. There are other ways in which you can get 85, 85, 86. e.g. 86 on first day, 85 on other two. Another way is 86 on second day and 85 on other two. So basically we are looking for the number of ways in which we can arrange 85, 85, 86 among 3 days. This can be done in 3!/2! = 3 ways. Probability of getting {85, 85, 86} is 1/27 * 3 = 1/9

Similarly, probability of getting {84, 85, 85} is also 1/9.

Probability of getting {84, 85, 86} will be 1/3 * 1/3 * 1/3 * 3! = 6/27

Total probability of getting median 85 = 1/27 + 1/9 + 1/9 + 6/27 = 13/27 _________________

That is what I got too.. but the answer is apparently : 13/27

Here is the OE..
First, lets figure out the different ways we could get a median of 85: 84, 85, 86; 84, 85, 85; 85, 85, 86; and 85, 85, 85. Now the probability of 84, 85, 86 is 1/3 * 1/3*/13 = 1/27 This arrangement could occur in 6 different ways (there are three days where the 84 could go, two where the 85 could go, and one for the 86: (3)(2)(1) = 6), giving us a probability of 1/27 * 6/1 = 2/9. Now the probability of 84, 85, 85 is 1/3, 1/3, 1/3 = 1/27 and there are 3 different ways to arrange the temperatures (the 84 could occur on three days, the other two days are 85), so the probability is 1/27 * 3/1 = 1/9. Similarly, the 85, 85, 86 is has a probability of 1/27 * 3/1 = 1/9. Finally, the probability of 85, 85, 85 is 1/3* 1/3*1/3 = 1/27 . To find the probability of a median of 85, we add up the probabilities of each of the scenarios: 2/9 + 1/9 + 1/9 + 1/27 . The correct answer is E.

Good question and equally good explanation, Karishma.....Thanks!!! But I think there was a spell-mistake in the explanation...

VeritasPrepKarishma wrote:

gk3.14 wrote:

The high temperature of a summer day on Fire Island is always one of three temperatures: 84, 85 or 86 degrees. If there is an equal chance of each temperature, what is the probability that the median high temperature over a three day period will be 85 degrees?

1) 1/27

2) 2/9

3) 1/3

4) 4/9

5) 13/27

When we are asked to find the median of 3 numbers, we arrange them in increasing order and then pick the middle number. The median of 85 will be obtained when the temperatures are one of: {85, 85, 85} {85, 86, 86}{85, 85, 86} {84, 85, 85} {84, 85, 86}

When will the temperatures be {85, 85, 85}? When every day the high is 85. What is the probability of this happening? 1/3 * 1/3 * 1/3 = 1/27

When will the temperatures be {85, 85, 86}? When the high is 85 on two days and 86 on another. What is the probability of this happening? Probability of 85 on first day is 1/3. Of 85 on second day is 1/3 and of 86 on third day is 1/3. Probability of this is 1/3 * 1/3 * 1/3 = 1/27. There are other ways in which you can get 85, 85, 86. e.g. 86 on first day, 85 on other two. Another way is 86 on second day and 85 on other two. So basically we are looking for the number of ways in which we can arrange 85, 85, 86 among 3 days. This can be done in 3!/2! = 3 ways. Probability of getting {85, 85, 86} is 1/27 * 3 = 1/9

Similarly, probability of getting {84, 85, 85} is also 1/9.

Probability of getting {84, 85, 86} will be 1/3 * 1/3 * 1/3 * 3! = 6/27

Total probability of getting median 85 = 1/27 + 1/9 + 1/9 + 6/27 = 13/27

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