Happens, or does not happen

The idea

The event 'not E', written Ē, is the complement of E; E and Ē are complementary events, and P(E) + P(Ē) = 1, so P(Ē) = 1 – P(E).

Two players, one winner

Sangeeta and Reshma play a tennis match. One of them must win. The probability that Sangeeta wins is 0.62.

"Reshma wins" is the same as "Sangeeta does not win".

What is the probability that Reshma wins?

• 0.62
• 0.38
• 1.62

Split the die into E and not E

Tap faces to put them in E. The faces left over make the event Ē ("not E"). Reach each target in order, and watch the two probabilities share the bar.

E and not E share the whole

P(E) + P(Ē) = 1 P(Ē) = 1 − P(E) The event "not E", written Ē, is the complement of E. E and Ē are complementary events.

Every outcome is in E or in Ē, never in both. So the two probabilities use up the whole bar. On a die, E = "a number greater than 4" has P(E) = 2/6 = 1/3, and Ē = "a number less than or equal to 4" has P(Ē) = 4/6 = 2/3. Check: 1/3 + 2/3 = 1.

It is a shortcut. When "not E" is easier to count, find P(Ē) and subtract it from 1. For two coins, "at least one head" is E. Its complement is "no head", the single outcome (T, T), so P(Ē) = 1/4 and P(E) = 1 − 1/4 = 3/4.

Notes

P(E) + P(Ē) = 1 P(Ē) = 1 − P(E) Ē ("not E") is the complement of E.

Check yourself

The probability that Sangeeta wins the match is 0.62. Find the probability that Reshma wins.

Answer: 0.38

P(R) = 1 − P(S) = 1 − 0.62 = 0.38.

One card is drawn from a well-shuffled deck of 52 cards. The probability that it is an ace is 1/13. Find the probability that it is not an ace. (Write a fraction, like 12/13.)

Answer: 12/13

1 − 1/13 = 13/13 − 1/13 = 12/13. (Check by counting: 48 cards are not aces, 48/52 = 12/13.)

Savita and Hamida are friends. The probability that they have different birthdays is 364/365 (ignore leap years). Find the probability that they have the same birthday. (Write a fraction, like 1/365.)

Answer: 1/365

1 − 364/365 = 365/365 − 364/365 = 1/365.

Two coins are tossed. To find P(at least one head) with the shortcut, which event do we take as the complement?

• no head, that is (T, T) — correct. Yes! (T, T) is exactly the outcome that is not in "at least one head". P(no head) = 1/4, so P(at least one head) = 1 − 1/4 = 3/4.
• at least one tail. This overlaps E: the outcome (H, T) has a head and a tail, so it is in both events. A complement shares no outcome with E.
• exactly one head. This leaves out (H, H), which is in neither event, so the two events would not use up the whole bar.