ProbabilityIntroduction
At the Diwali mela, Riya runs a small game stall. She shakes two dice in a brass cup, turns it upside down on the table and peeks underneath. Her friend Aman has to bet on just one thing: is it a double?
With no information, Aman knows the answer from Class 11: 6 of the 36 equally likely outcomes are doubles, so the chance is . But Riya is a generous host. Before Aman bets, she gives him one clue about the dice.
Play Aman. Tap a clue: every outcome the clue rules out fades away, and the chance of a double is worked out again on what is left. Complete all three missions.
A clue does not change the dice — they are already lying under the cup. It changes what you know, and so it changes which outcomes are still possible. "The sum is 10 or more" leaves only 6 outcomes, and 2 of them are doubles, so the chance jumps from to . "The sum is odd" leaves no double at all. And "the first die shows 3" leaves the chance exactly where it was.
Probability With Information
In Class 11 you met the
This chapter asks a new question, the one Riya's stall just asked: how does the probability of an event change once we know that another event has happened? That idea, conditional probability, opens three more doors:
- the multiplication rule, for events that happen one after another,
- independence, when the information turns out to be useless, and
- Bayes' theorem, which runs the reasoning backwards, from an effect you have seen to the cause that probably produced it.
Throughout the chapter, the outcomes of an experiment are equally likely unless we say otherwise.
A quick warm-up. One die is thrown, and you are told that the number is even. Which outcomes are still possible?
Right — the information shrinks the sample space from six outcomes to three. Hold on to that picture: it is the whole of the next section.