Chance Encounters

Chance Encounters

Probability turns "probably", "unlikely" and "fifty-fifty" into numbers you can calculate with. A probability is always between 0 and 1: • 0 = impossible, 1 = certain, 0.5 = an even chance. • The closer to 1, the more likely; the closer to 0, the less likely. Write probabilities as fractions, decimals or percentages: never as a ratio like 3 : 5. A ratio scores no marks for a probability.

Match the word to its probability

Theoretical probability

When outcomes are equally likely, you can calculate the probability by counting: P(event) = favourable outcomes ÷ total outcomes. A fair six-sided die has 6 equally likely outcomes. Three of them are even (2, 4, 6), so P(even) = 3 ÷ 6 = 1/2 = 0.5.

Spin the spinner

A fair spinner has 5 equal sections: 2 are red and 3 are blue. What is the probability of landing on RED? Give your answer as a decimal.

Everything sums to 1

The probabilities of all possible outcomes always add up to 1. That gives a powerful shortcut for "not" questions: P(not A) = 1 − P(A). If the probability it rains tomorrow is 0.3, then the probability it does NOT rain is 1 − 0.3 = 0.7.

The complement

The probability that a spinner lands on red is 0.45. What is the probability it does NOT land on red? Give your answer as a decimal.

Mutually exclusive events

A bag holds red, blue and green counters. P(red) = 0.4 and P(blue) = 0.35. A counter cannot be two colours at once, so what is P(red OR blue)?

• 0.75
• 0.14
• 0.05
• 0.25

Experiments and expected frequency

Sometimes outcomes are not equally likely, such as a bent coin or a drawing pin, so you experiment instead of counting. Three terms here get mixed up with one another:

Expected frequency

The probability a biased coin lands heads is 0.3. If it is flipped 40 times, how many heads would you EXPECT? (Give a whole number.)

Which estimate is more reliable?

Ann throws a drawing pin 20 times to estimate P(point up); Ben throws it 500 times. Whose relative frequency is likely to be the more reliable estimate?

• Ben: more trials means the relative frequency is closer to the true probability
• Ann: fewer throws gives a cleaner result
• They are equally reliable
• Neither can be used as an estimate

Pick the rule

Four questions, no headings. For each, choose the rule you would reach for first.

• A bag holds 5 red and 3 blue counters, all equally likely to be picked. Find P(red).
• The probability that a train is late is 0.15. Find the probability that it is NOT late.
• A spinner gives P(red) = 0.2 and P(green) = 0.45, and it cannot land on both. Find P(red or green).
• A biased dice lands on six with probability 0.3. How many sixes would you expect in 200 rolls?

Probability rules

On the probability scale, an impossible event has probability _____ and a certain event has probability 1. Because all outcomes sum to 1, the probability an event does NOT happen is found by _____ the probability it does from 1. To predict results, expected frequency = probability _____ the number of trials.