Probability flashcards

Topic 8: Probability

The probability scale

0 means impossible and 1 means certain. Every probability lies from 0 to 1. Give it as a fraction, decimal or percentage, never as a ratio like 5:2.

Probability that A does not happen

P(A′) = 1 − P(A)

Relative frequency

Times the event happened ÷ number of trials. It is an estimate of the probability and improves as the number of trials grows.

Expected frequency: a pin lands point up with relative frequency 0.3. How many times in 1500 trials?

Expected frequency = probability × number of trials = 0.3 × 1500 = 450.

Fair and biased

Fair: all outcomes are equally likely. Biased: outcomes are not equally likely; some are favoured.

How do you find P(at least one)?

1. Write the opposite event, "none". 2. Find P(none). 3. P(at least one) = 1 − P(none).

Tree diagrams: when do you multiply and when do you add?

Multiply along the branches of one path. Add the probabilities of different paths that give the event. Branches from one point add up to 1.

P(test) = 0.4. If there is a test, P(revise) = 0.9; if not, P(revise) = 0.5. Find P(revise).

0.4 × 0.9 + 0.6 × 0.5 = 0.36 + 0.30 = 0.66

A bag has 4 red and 6 blue counters. Two are taken without replacement. Find P(both red).

4/10 × 3/9 = 12/90 = 2/15. Without replacement, the second denominator drops by one.

Same bag (4 red, 6 blue), without replacement. Find P(one of each colour).

Two paths: 4/10 × 6/9 + 6/10 × 4/9 = 24/90 + 24/90 = 48/90 = 8/15.

Two fair spinners numbered 1 to 4 are spun and the scores added. Find P(total = 4).

Sample space has 4 × 4 = 16 outcomes. (1, 3), (2, 2), (3, 1) give 4. P = 3/16

Of 40 people, 18 speak French, 14 speak German and 12 speak neither. How many speak both?

At least one language: 40 − 12 = 28. 18 + 14 − both = 28, so both = 4.