Flipping Your Chances

Probability Level 2

Bobby has a coin, a standard US 2007 penny. (That's not important to the question.) His friend Billy bets him 1 dollar that he can't flip it and get heads twice in a row. Bobby knows that the probability of flipping a coin and getting heads twice in a row is $\frac{1}{4}$ . So Bobby takes the bet.

Bobby flips the coin once and gets heads. Then Bobby says he's going to go "get some water." (Did he really do that? I don't know.) 10 minutes later, (Hmmm... probably didn't just get some water if he was gone for 10 minutes...) Bobby comes back and flips the coin again. And Billy looks down, examines the coin, in absolute shock. Bobby is, however, not surprised by his results. In fact, he is very proud of his result.

Billy wonders if there was something Bobby did to make sure he got the second flip to come up heads. "Why did Bobby leave for 10 minutes?" Billy wondered. He knew that getting two heads in a row has a chance of $\frac{1}{4}$ . He also knew that two independent actions did not multiply the chances. For example, if Bobby and Billy flip a coin, the probability for Bobby and Billy to get both heads is still $\frac{1}{2}$ . The two actions do not affect each other.

But Billy wondered if Bobby leaving for 10 minutes "reset" the chances and put it back at $\frac{1}{2}$ . Could it be so?

Did Bobby leaving for 10 minutes to "get some water" somehow increase the chances of Bobby getting heads twice in a row?

Yes No

1 solution

Jacob Day
Oct 18, 2017

This one is kind of easy. Billy suspects Bobby "reset" the chances of him getting two heads in a row. However, that is not the case. Bobby actually still only had a chance of $\frac{1}{4}$ to get heads twice in a row. No coin flips were done in the 10 minutes that Bobby was gone.

The reason that the chance of getting two heads in a row is $\frac{1}{4}$ is that these coin flips are independent actions. That means one caused the other. You can't get two heads in a row when two separate people do them, like the Bobby and Billy example. The probability of getting two heads in a row is as follows:

$P =$ $\frac{1}{2}$ $\times$ $\frac{1}{2}$

$\frac{1}{2}$ - for the first coin flip $\frac{1}{2}$ - for the second coin flip

Please make sure to tell me if any of information is wrong.

Actually, they are independent actions. This is why you are able to multiply them and get 1/4. The reason getting water does not work is because getting water is also independent of coin flips.

Siva Budaraju - 3 years, 7 months ago

Thanks for the correction.

Jacob Day - 3 years, 7 months ago

Thanks for correcting it, but some information is still wrong. Actually, independent actions mean neither caused the other; you can get two heads in a row when 2 separate people flip it, as the flips are still independent events.

Siva Budaraju - 3 years, 7 months ago

Flipping Your Chances

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