Question

Mike and Jose flip the same coin every day at work to decide who pays for the morning coffee. After looking over credit card receipts, Jose realizes that he's paid 73 of the last 100 days. Can Jose conclude that there's something funny about the coin they're flipping,explain the reasoning?

Answer 1

First of all, let us understand the basic two approaches behind the statistics and probability.

Let us declare the event A to be turning up of heads (Jose paying the bill), whenever the coin is tossed.

To understand probabilty, we shall know there are two approaches towards probability:

1) Classiscal Approach

Under classical approach, the probability of event A happening is simply known by the formlula = No. of favorable cases / Total number of outcomes

Therefore, under this approach, we get, P(A)= 1/2. (assuming an unbiased coin)

(understand that equilikeliness of the events associated with experiment is assumed to be true here, ie,. if there are 2 outcomes of an experiment, then both have a probability 0.5 of happening)

2) Mathematical Approach

This is a better approach that does not assume the equilikeliness of events, but rather tends to approach the equilikeliness of the events.

It is defined as the number of trials of an experiment tend towards infinity, the limiting form of the below expression tends to approach towards classical approach of probability:

limit ( n tending to infinity) { Number of times A comes up / Total number of trials of experiment} ----> Classical Approach.

Now coming to the given problem, we can have two point of views with respect to these approaches:

1) Classical Approach

As in the case of any unbiased coin, P(A)= 0.5.

But as it turns out to be, after successive trials of an experiment, P(A) comes out to be 0.73. This literally depicts that the coin is a biased one in favor of Mike, ie, Probability of getting a tail (Mike paying the bill) is 1-0.73= 0.27. Hence, he has a pretty greater chance of not paying the bill. If it had been an unbiased one, then they would have paid the bill equal number of times.

2) Mathematical Approach

As we the number of trials are tending towards a large number, 100, the probability should have approached classical approach, ie, it should have been approximated to 50 times Mike paying the bill and 50 times Jose paying the bill. But as of now, P(A= jose paying the bill) = 0.73= 73/100, which is not tending towards classical approach with unbiased coin.

Therefore, it is clear that the coin being used is literally biased one, and has to be changed so that they can settle the same.