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Conditional probability
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What is the probability of getting two sixes in a row when rolling one six-sided dice?
Now we're going to roll a die. If the die shows a five or a six, you win. If it shows a four or a smaller number, then I win. What is the probability that you win? Do you remember?
The number of outcomes favourable for you divided by the number of all possible outcomes. The probability of your winning is one third. Okay, you won. But what happens if we use two dice? What's the probability that both dice show a six?
These are all the possible outcomes. Thirty-six of them. And this is the only favourable outcome. One divided by thirty-six. Zero comma zero-two-eight.
Another way to show what's happening is to draw a tree diagram. First we draw all possible outcomes for one die. Then for each one of these outcomes we draw all possible outcomes for the other die. 36 different paths, one of which leads to two sixes. The probability of the second die showing a six, given that the first die also shows a six, is called conditional probability.
Look at the tree diagram. The probability of rolling a six with the first dice is one sixth. And if the first die already shows a six, then there is one out of six chances to get it on the other die too. So we have a one sixth of one sixth chance of rolling two sixes. And we write this as: One sixth times one sixth.
One thirty-sixth! Let's take another, a bit harder problem: How large is the probability of rolling a six on exactly one of the two dice? The tree diagram looks like this: Roll a six on the first die, but not on the second. Or don't roll a six on the first, but do on the second. There are ten outcomes out of 36 possible, or just under a 28% chance.
To calculate this you need to do the following: The probability of rolling a six on the first die, times the probability of not rolling a six on the second die. Plus: The probability of not rolling a six on the first die, times the probability of rolling a six on the second die. Ten out of thirty-six. The probability does not predict what will happen, just how probable it is for something to happen. Feel free to experiment with dice yourself and compare your calculations.