Between 0 and 1

The idea

Since the number of favourable outcomes is always less than or equal to the number of all possible outcomes, every event E satisfies 0 ≤ P(E) ≤ 1.

Two answers for one die

Two students find the probability of an event on a fair die. Asha gets 7/6. Bela gets −1/6.

Who can be right?

• Asha only
• Bela only
• Neither of them

Paint tiles and watch the marker

Two dice, one blue and one grey, make 36 outcomes. Each tile is one outcome. Paint the tiles that are favourable to your event E (tap, or drag across tiles) and watch P(E) on the line. Reach each target in order. Then try to push the marker past the ends.

Never below 0, never above 1

0 ≤ P(E) ≤ 1 for every event E.

The number of favourable outcomes can never be more than the number of all possible outcomes, so the top of the fraction is never more than the bottom. It cannot be less than 0 either, since you cannot have fewer than none. So favourable / possible always lies from 0 to 1.

The two ends are the two special events you met: 0 is the impossible event and 1 is the sure event. Every other event lies in between. Asha's 7/6 is more than 1 and Bela's −1/6 is below 0, so neither can be a probability.

Notes

0 ≤ P(E) ≤ 1 for every event E, because favourable outcomes never outnumber possible outcomes.

Check yourself

Which of these can not be the probability of an event?

Why can a probability never be more than 1?

On a die, an event E has x favourable faces, so P(E) = x/6. How many different whole-number values can x take?

Answer: 7

x can be 0, 1, 2, 3, 4, 5 or 6: that is 7 values, and the probabilities 0/6, 1/6, …, 6/6 all lie from 0 to 1.

We know 0 ≤ P(E) ≤ 1. What can we say about P(Ē) = 1 − P(E)?

• 0. This can be a probability: it is the probability of an impossible event.
• 7/12. This lies between 0 and 1, so it can be a probability.
• 3/2 — correct. Yes! 3/2 is more than 1. Favourable outcomes can never outnumber the possible outcomes.
• Because the favourable outcomes are some of the possible outcomes, so there cannot be more of them than all of them — correct. Yes! Favourable ≤ possible, so favourable / possible is at most 1.
• Because a die has only six faces. This holds for every experiment, not only for dice. The reason is about favourable and possible outcomes.
• Because a probability must be a fraction like 1/2. A probability can be exactly 1 (a sure event) or 0 (an impossible event). The reason is that favourable outcomes never outnumber possible outcomes.
• It can be negative. P(E) is at most 1, so 1 − P(E) is at least 0. It is never negative.
• It also lies from 0 to 1 — correct. Yes! If P(E) is between 0 and 1, then 1 − P(E) is also between 0 and 1. (Ē is an event too.)
• It can be more than 1. P(E) is at least 0, so 1 − P(E) is at most 1.