# How to determine a probability?

A coin is tossed and a head or a tail observed. If a head results, the coin is tossed second time. If a tail results on the first toss, a die is rolled.

a) Draw a tree diagram and list the sample space for this experiment.

b) What is the probability that a die is rolled in the second stage of this experiment

The answer in the picture below is the answer for questionA. If any of you can help with the questionB because I am a bit confused?

• If I understand well then a die is rolled in the second stage if and only if a tail is observed in the first stage. What is the probability that indeed a tail is observed in the first stage? – drhab Sep 30 '14 at 8:17

## 3 Answers

There is a very classical understanding of probability where outcomes are simply counted. In that case, you might think the answer is 6/8 because six of the eight outcomes involve flipping T on the coin.

However, it seems unlikely that the probability of flipping a coin and getting tails is really 3/4. Assuming the coin is "fair," the probability of getting tails is 1/2 (and likewise, heads is also 1/2, since "fair" means the probabilities are equal).

You see, you cannot simply count six out of eight outcomes unless all the outcomes are equally likely (they are not -- some have probability 1/4, some 1/12). Even though the coin is "fair" and has equally likely outcomes, just as the die does, the probabilities are different (1/2 for the coin, 1/6 for the die).

But the answer is simple, you roll the die exactly when you flip the coin and get tails on the first flip -- which happens with probability 1/2.

You can also compute the probabilities of each event, see attached. You'll notice, not only do you see 1/2 leading up to the first T flip (green), but the sum of probabilities of all possible outcomes that involve a die being rolled (blue) is also:

1/12 + 1/12 + 1/12 + 1/12 + 1/12 + 1/12 = 1/2

which agrees with the 1/2 we had from looking to the green node.

• so, from the attachment above, the probability is the 1/4 and 1/12?isn't it? – Agus Maloco Sep 30 '14 at 11:45
• All of the individual outcomes have probability 1/4 or 1/12. – Kellen Myers Sep 30 '14 at 15:26

The likelihood of rolling the die, is conditional on the probable outcomes of the first event.

What is the probability for the first event coming up tails? And therefore, how likely are you to roll a die in the next event?

Hint: at every node of your tree, the branches linked to the node have the same probabilities. Hence your probabilities $6/8$ and $2/8$ are wrong.