Look up!

Probability Level 4

Four people play a game wherein they stand in a circle (or in this case four corners of a square) all look down, and then they all look up at the same time and into one of the other three people's eyes randomly.

If the probability that at least two people will look at each other is $\dfrac{a}{b}$ , where $a$ and $b$ are coprime positive integers, what is $a+b$ ?

Image credit: http://bradhallart.blogspot.com/

The answer is 44.

1 solution

Geoff Pilling
Oct 7, 2016

The total number of ways that at least two people look at each other is derived as follows:

1) Pick two people to look at each other (6 ways)
2) Multiply by 9 (9 ways for the remaining two people to look)
3) Subtract 3 (3 possibilities have been double counted above, where 2 pairs look at each other)

So, the total number of ways they can look at each other is:

$6*9-3 = 51$

And the total number of ways the group can look at each other is $3^4 = 81$ (since each person can look 3 different ways).

So, the probability that there will be at least one pair looking at each other is given by $\frac{51}{81} = \frac{17}{27}$ .

$17+27 = \boxed{44}$

Hmmm i made a wrong assumption. I thought they were all in 1 line. But thats not the case since they look on of the 3 in the eyes.

Peter van der Linden - 4 years, 8 months ago

Good point tho... Lemme clarify the question...

Geoff Pilling - 4 years, 8 months ago

I made an even more stupid assumption. I thought that the only case where there are not two mutually looking people is when they all form a loop.

William Nathanael Supriadi - 4 years, 8 months ago

Haha, yep... In fact, as I'm sure you figured out, there are two such situations:

The one you mention, a loop of 4
A loop of three, with the fourth looking at any one of the other three

Hope you enjoyed the problem!

Geoff Pilling - 4 years, 8 months ago

Look up!

Image credit: http://bradhallart.blogspot.com/

The answer is 44.

1 solution

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