Question 1203341: Topic : Discrete Random Variable (Geometric Distribution)
One in four boxes of Zingo breakfast cereal contains a free toy. Let the random variable X be the number of boxes that a child opens, up to and including the one in which they find out their first toy.
a) Find the mode and the expectation of X
b) Interpret the two values found in part (a) in the context of this question
Answer by ikleyn(52754)   (Show Source):
You can put this solution on YOUR website! .
Topic : Discrete Random Variable (Geometric Distribution)
One in four boxes of Zingo breakfast cereal contains a free toy.
Let the random variable X be the number of boxes that a child opens,
up to and including the one in which they find out their first toy.
a) Find the mode and the expectation of X
b) Interpret the two values found in part (a) in the context of this question
~~~~~~~~~~~~~~~~
The random variable X may have these values: 1, 2, 3 and 4.
It may not have other values.
If X= 1, it means that the secret toy was found in the 1st opened box.
The probability of this event is  , obviously.
So, P(X=1) = .
If X= 2, it means that the secret toy was found in the 2nd opened box,
while the first box was "empty".
The probability of this event, is  = , obviously.
So, P(X=2) = .
If X= 3, it means that the secret toy was found in the 3rd opened box,
while the boxes 1 and 2 were "empty".
The probability of this event, is  = .
So, P(X=3) = .
If X= 4, it means that the secret toy was found in the 4th opened box,
while the boxes 1, 2 and 3 were "empty".
The probability of this event, is  = .
So, P(X=4) = .
Thus we have found analytically this remarkable and unexpected ANSWER:
+----------------------------------------------------+
| the random variable X is UNIFORMLY DISTRIBUTED |
| on the set of numbers {1, 2, 3, 4}. |
+----------------------------------------------------+
But, from the other point of view, it seems to be natural: what is the difference
between the boxes #1, #2, #3 and #4 ? - - - There is no any difference - they all are identical (!)
The mathematical expectation of this random variable X is
ME =  +  +  +  =  =  = 2 = 2.5.
It means, that in AVERAFGE, 2.5 boxes should be opened to find the secret toy,
if the experiment is repeated many times.
Notice that this number 2.5 lies in the central point on the number line
for this set of numbers {1, 2, 3, 4}.
Solved.
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