Probability: Quick Revision

A brief note on Probability. Try each question at the end before you open its answer.

The basics

• A probability is a number from $0$, impossible, to $1$, certain.
• With equally likely outcomes: $P = \dfrac{\text{outcomes we want}}{\text{all outcomes}}$.
• Not: $P(\text{not } A) = 1 - P(A)$.

When we cannot count

• Relative frequency = successes divided by tries. It estimates the probability.
• A short run jumps about; a long run settles near the true probability.
• A coin or a die has no memory: past results do not change the next chance.

Two events together

A worked example

A coin is tossed and a die is rolled. What is the chance of a head and a 5 or more?

1. All the outcomes. $2 \times 6 = 12$ equally likely pairs.
2. The ones we want. (H, 5) and (H, 6): $2$ pairs.
3. The chance. $\dfrac{2}{12} = \dfrac{1}{6}$.
4. A check. The events do not affect each other, so multiply: $\dfrac{1}{2} \times \dfrac{2}{6} = \dfrac{1}{6}$.

Practice with new numbers

A program to try

The program lists all 12 outcomes of the worked example and counts the ones we want.

Mistakes to avoid

• A probability above 1 or below 0. It cannot be.
• Treating the 11 sums of two dice as equally likely. The 36 pairs are; the sums are not.
• Adding for "both". Multiply the chances of events that do not affect each other.
• Forgetting that the bag changes when the first item is not put back.

Check yourself

1. A letter is picked at random from the word STATISTICS. What is the probability that it is a T?

2. The chance of a delay is $0.12$. What is the chance of no delay?

3. A cup lands upright 54 times in 150 throws. About how many upright landings do you expect in 500 throws?

4. Two cards are drawn from five cards numbered 1 to 5, without putting the first back. What is the chance that both are odd?