Probability flashcards

Topic 5: Probability

Probability of an event not happening

P(not A) = 1 − P(A), because exhaustive outcomes sum to 1.

Relative frequency and expected frequency

Relative frequency = times the outcome happened ÷ number of trials. Expected frequency = probability × number of trials. More trials give a more reliable estimate.

Mutually exclusive compared with independent events

Mutually exclusive: cannot happen at the same time, so P(A and B) = 0. Independent: one does not change the probability of the other, so P(A and B) = P(A) × P(B).

The "or" rule

P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events, P(A or B) = P(A) + P(B).

How do you use a tree diagram?

1. Each pair of branches from one point sums to 1. 2. Multiply along the branches for each outcome. 3. Add the outcomes that fit the event. Without replacement, the second-pick denominators drop by 1.

3 green and 2 white beads; two are taken without replacement. P(both white)?

2/5 × 1/4 = 2/20 = 1/10

Two fair dice are rolled. P(the sum is 9)?

There are 36 equally likely outcomes. Sum 9: (3,6), (4,5), (5,4), (6,3), so 4/36 = 1/9.

How do you work out P(at least one)?

1 − P(none).

Conditional probability formulaHigher

P(A given B) = P(A and B) ÷ P(B). "Given" cuts the sample space down to B: in a two-way table or Venn diagram, divide by the B total, not the grand total.

60% of pupils take the bus and 25% of bus users are late. 20% of all pupils are late. P(took the bus given late)?Higher

Out of 100 pupils: 60 take the bus, 15 of those are late, 20 are late in total. P(bus given late) = 15/20 = 0.75