The high temperature of a summer day on Fire Island is always one of

Question

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

A. 1/27

B. 2/9

C. 1/3

D. 4/9

E. 13/27

KarishmaB · Accepted Answer

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

XxxyyY · Answer

1/3?

84-85-86, lets simplify --> 4-5-6

There are 27 possible combinations, 9 out of 27 can have 5 as median.
9/27 --> 1/3

EconGirl · Answer

probailities:
day 1 (any temperature) = 3/3
day 2 (85 only) = 1/3
day 1 (any temperature) = 3/3

so 3/3 x 1/3 x 3/3 =9/27 = 1/3

agsfaltex · Answer

Total number of ways:P 3*3*3 = 27

Possible  permutations  as 85 median

(85, 85, 85), or any  permutation  of all three.

Total number of permutations: 3 + (3*2*1) = 9

Therefore, 9/27 = 1/3

gk3.14 · Answer

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.

Viperace · Answer

Hmm, we missed the fact that

median of (4,5,5),(6,5,5)... is 5 

A good lesson

Chetangupta · Answer

Can some expert explain this question pls?

krishp84 · Answer

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

ScottTargetTestPrep · Answer

Since each day has 3 possible outcomes for the high temperature, 3 consecutive days have 3 x 3 x 3 = 27 total possible outcomes for the high temperature. The number of ways to get 85 as a median can occur in the following ways:

One reading of 84, two readings of 85: By the permutations of indistinguishable objects formula, this can occur in 3!/2! = 3 ways. (which are (84, 84, 85), (84, 85, 84); (85, 84, 84)).

One reading of 86, two readings of 85: This can also occur in 3 ways, as explained above.

One reading of each temperature: This can occur in 3! = 6 ways.

Three readings of 85: This can occur in one way.

Thus, a median high temperature of 85 degrees occurs in 3 + 3 + 6 + 1 = 13 outcomes. Thus, the probability is 13/27.

Answer: E

gvij2017 · Answer

I want to clear a doubt here.

Question ask that the median high temperature over three days will be 85. So should keep days temperature in order of day 
 or I can again rearrange the temperatures.

As you did.

I agree that (84,85, 86) can be arranged in 6 ways. But if that order matters to me then how in some case 85 can be median. For example,

order of temperature is 86 84, 85, then without rearrangement how median could be 85. And how could I consider such case for calculating the probability. 
In your case you have considered rearrangement of temperatures.

Please explain.

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