All the pieces make one whole
The probabilities of all the elementary events add up to 1.
Add up every elementary event
A coin has 2 elementary events (head, tail). A bag with a red, a blue and a yellow ball has 3. A die has 6. In each experiment all the elementary events are equally likely.
Now add up the probabilities of all the elementary events of an experiment.
What total do you expect?
What this lesson covers
The idea
The sum of the probabilities of all the elementary events of an experiment is 1.
Add up every elementary event
A coin has 2 elementary events (head, tail). A bag with a red, a blue and a yellow ball has 3. A die has 6. In each experiment all the elementary events are equally likely.
Now add up the probabilities of all the elementary events of an experiment.
What total do you expect?
- Always 1
- The number of elementary events: 2, 3, 6
- Always 1/2
Fill the bar
Every outcome is an elementary event with the same probability. Tap an outcome to pour its probability into the bar. The whole bar is 1. Fill the bar for the coin, the bag and the die.
The slices fill the whole bar
The sum of the probabilities of all the elementary events of an experiment is 1.
Coin: 1/2 + 1/2 = 1 Bag of three balls: 3 × 1/3 = 1 Die: 6 × 1/6 = 1
Every outcome gets its own slice of the bar, and all the slices together fill it exactly.
NCERT checks this in Example 2: P(yellow) + P(red) + P(blue) = 1.
Notes
The sum of the probabilities of all the elementary events of an experiment is 1.
Check yourself
Two coins are tossed. The four elementary events (H, H), (H, T), (T, H), (T, T) have probability 1/4 each. Find the sum of the probabilities of (H, H), (H, T) and (T, H). (Write a fraction, like 3/4.)
Answer: 3/4
1/4 + 1/4 + 1/4 = 3/4. The missing slice, P(T, T) = 1/4, would make the total 1.
Which could be the probabilities of the three elementary events of an experiment?
An experiment has four elementary events. Three of them have probabilities 1/2, 1/4 and 1/8. Find the probability of the fourth. (Write a fraction, like 1/8.)
Answer: 1/8
The three add up to 7/8, so the fourth is 1 − 7/8 = 1/8.
One card is drawn from a well-shuffled deck of 52 cards, so each card is an elementary event with probability 1/52. What is the sum of the probabilities of all 52 elementary events?
Answer: 1
52 × 1/52 = 52/52 = 1.
- 1/2, 1/3, 1/3. The total is 1/2 + 1/3 + 1/3 = 7/6, which is more than 1. The total must be exactly 1.
- 1/4, 1/4, 1/4. The total is 3/4. A slice of 1/4 is missing, so these cannot be all the elementary events.
- 1/2, 1/3, 1/6 — correct. Yes! 1/2 + 1/3 + 1/6 = 6/6 = 1.