Too many outcomes to count

The idea

The classical definition needs a finite number of outcomes; when equally likely outcomes fill a length or a region, probability is the favourable length (or area) divided by the total length (or area).

The musical chair game

In a musical chair game, the person playing the music may stop it at any moment within 2 minutes after she starts.

The stopping moment can be any number between 0 and 2: 0.3 minutes, 1.07 minutes, 1.999 minutes, and so on.

Can you list and count all the possible outcomes?

• Yes, there are 2: one for each minute
• Yes, there are 120: one for each second
• No, there are infinitely many moments

Measure instead of counting

Music. Set the end of the window to the first half-minute (0 to 1/2) and stop the music 100 times. Helicopter. A helicopter crashed somewhere in a region; search it 100 times. Compare each relative frequency with the ratio of lengths or areas.

Length over length, area over area

When equally likely outcomes fill a length or a region, P(E) = favourable length / total length, or favourable area / total area.

Music: the favourable window is from 0 to 1/2, a length of 1/2. The total length is 2. So P(E) = (1/2)/2 = 1/4.

Helicopter: the region is 4.5 × 9 = 40.5 km² and the lake is 2.5 × 3 = 7.5 km². So P(crashed in the lake) = 7.5/40.5 = 5/27.

NCERT marks Examples 10 and 11 as "not from the examination point of view". They show why counting outcomes fails here, and what to use instead.

Notes

When equally likely outcomes fill a length or a region, P(E) = favourable length / total length, or favourable area / total area.

Check yourself

The music stops at any time within 2 minutes. Find the probability that it stops in the first minute. (Write a fraction, like 1/2.)

Answer: 1/2

Favourable length 1, total length 2: P = 1/2.

The music stops at any time within 2 minutes. Find the probability that it stops between 0.5 and 1.25 minutes after the start. (Write a fraction like 3/8 or a decimal.)

Answer: 3/8

Favourable length = 1.25 − 0.5 = 0.75. Total length = 2. P = 0.75/2 = 0.375 = 3/8.

In the helicopter region (9 km by 4.5 km, area 40.5 km²), a forest measures 4.5 km by 3 km. Find the probability that the helicopter crashed in the forest. (Write a fraction, like 1/3.)

Answer: 1/3

Forest area = 4.5 × 3 = 13.5 km². P = 13.5/40.5 = 1/3.

Why can we not find "the music stops in the first half-minute" by counting favourable outcomes and all possible outcomes?

• Because the music might never stop. The problem says it stops within 2 minutes. The trouble is that there are too many possible moments to count.
• There are infinitely many possible stopping moments, so they cannot be counted — correct. Yes! Between 0 and 2 there are infinitely many numbers, so favourable / possible has nothing to divide by. We use lengths instead.
• Because the moments are not equally likely. We assume the moments are equally likely. The trouble is only that there are infinitely many of them.