Count the good ones

The idea

The theoretical (classical) probability of an event E is P(E) = Number of outcomes favourable to E / Number of all possible outcomes, assuming that all the outcomes are equally likely.

A number greater than 4

A fair die is thrown once. Let E be the event getting a number greater than 4. The chance of E is called its probability, written P(E).

The die has 6 faces. Some of them make E happen.

What do you think P(E) is?

• 1/6
• 1/3
• 1/2

Count the faces, then throw the die

First tap the faces that make E happen. The toy shows the count. Then throw the die many times: it counts how often E happened and shows the relative frequency (times E happened ÷ number of throws).

Favourable out of possible

P(E) = favourable outcomes / possible outcomes, when all the outcomes are equally likely. It is the theoretical (classical) probability.

Favourable outcomes are the ones that make E happen: here 5 and 6, so 2 of them. Possible outcomes are all the faces: 6. So P(E) = 2/6 = 1/3.

We found P(E) by counting, without throwing the die even once. The relative frequency from real throws (the experimental probability you met in Class IX) wobbles at first, and then settles near the theoretical probability as the number of throws grows.

Pierre Simon Laplace gave this definition in 1795.

Notes

P(E) = favourable outcomes / possible outcomes, when all outcomes are equally likely. The relative frequency of many throws settles near it.

Check yourself

A box has 3 blue, 2 white and 4 red marbles. One marble is drawn at random. Find the probability that it is white. (Write a fraction, like 2/9.)

Answer: 2/9

There are 2 white marbles out of 3 + 2 + 4 = 9 marbles, so P(white) = 2/9.

A class has 40 students, 25 girls and 15 boys. The teacher draws one name card at random. Find the probability that it is a girl. (Write a fraction, like 5/8.)

Answer: 5/8

25 girl cards out of 40 cards: P(girl) = 25/40 = 5/8.

A carton has 100 shirts: 88 are good, 8 have minor defects and 4 have major defects. Sujatha rejects only the shirts with major defects. One shirt is drawn at random. Find the probability that it is acceptable to Sujatha.

Answer: 0.96

She accepts 88 + 8 = 96 shirts out of 100, so the probability is 96/100 = 0.96.

On a fair die, P(getting a 5) = 1/6. What does this mean?

• In every 6 throws, a 5 shows up exactly once. No. A probability is not a promise for a few throws. A 5 can show up 0 times or 3 times in 6 throws.
• A 5 is less likely than the other faces. No. All 6 faces are equally likely, so every face has probability 1/6.
• 1 of the 6 equally likely faces is favourable, and in very many throws a 5 shows up about 1/6 of the time — correct. Yes! Favourable 1, possible 6. And as the throws increase, the relative frequency settles near 1/6, as you saw.