8. Probability & Statistics

Lesson

Calculating probabilities

We can calculate the theoretical probability of an event by constructing a fraction like this: $\frac{\text{what you want}}{\text{total }}$what you wanttotal which we write more formally as

P(event) = $\frac{\text{total favorable outcomes }}{\text{total possible outcomes }}$total favorable outcomes total possible outcomes

An expected outcome, as the name suggests, is the value (probability value) we expect from a given sample space or a set of given individual event probabilities.

The best way to describe this might be through an example.

A spinner is divided equally into $8$8 sections, but $3$3 of them are colored green.

**a)** What is the probability of landing on green?

$\frac{\text{total favorable outcomes }}{\text{total possible outcomes }}=\frac{3}{8}$total favorable outcomes total possible outcomes =38

**b)** If the spinner is spun $145$145 times, how many times would you expect it to land on green?

We take the probability of the event P(green) and multiply it by the number of trials.

P(green) x $145$145 = $\frac{3}{8}\times145=54.375$38×145=54.375

At this point we need to round appropriately, so we could say that if the spinner is spun $145$145 times we could expect it to be green $54$54 times.

A car manufacturer found that $1$1 in every $4$4 cars they were producing had faulty brake systems. If they had already sold $5060$5060 cars, how many of those already sold would need to be recalled and repaired?

Data collected in a certain town suggests that the probabilities of there being $0$0, $1$1, $2$2, $3$3, $4$4 or $5$5 or more car thefts in one day are as given in the table below.

What is the expected number of car thefts occurring on any particular day (to two decimal places)?

Car Thefts | $0$0 | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 or more |

Probability | $0.09$0.09 | $0.24$0.24 | $0.20$0.20 | $0.16$0.16 | $0.18$0.18 | $0.13$0.13 |

$260$260 standard six-sided dice are rolled.

What is the probability of getting an even number on a single roll of a die?

How many times would you expect an even number to come up on the $260$260 dice?

We have already learned about probability which is the chance of an event occurring. However, if you have ever done a probability activity in class, you may have noticed that what you **expected** to happen was different to what **actually** happened. In math, we call this the expected frequency and the experimental (or observed) frequency. Remember frequency means how often an event occurs.

The expected frequency is how often we think an event will happen. For example, when we flip a coin, we would expect it to land on heads half the time. So if we flipped a coin $100$100 times, we would expect it to land on heads $50$50 times (because $\frac{50}{100}=\frac{1}{2}$50100=12).

The experimental or observed frequency is the number of times an event occurs when we run an experiment. For example, let's say we actually decided to flip a coin $100$100 times. We expected tails to come up $50$50 times but it only happened $47$47 times. We would say that the experimental frequency for getting a tail is $\frac{47}{100}$47100. Did you notice that the experimental frequency was different to the observed frequency.

Now it's your turn to compare expected and experimental frequencies.

- Before you start playing with the applet, write down how often you would expect the spinner to land on each color (i.e. the expected frequency for each color). Discuss your answers in groups.
- Spin that spinner! The table will record the experimental frequencies. Discuss why the probabilities change after each spin.
- After you have finished the experiment, compare the expected frequencies to the experimental frequencies. What did you notice?

What is the probability of rolling a 3 on a standard die?

A coin was flipped $178$178 times with $93$93 tails recorded.

What is the exact experimental probability of flipping tails with this coin?

Develop a probability model and use it to find probabilities of events. Compare experimental and theoretical probabilities of events. If the probabilities are not close, explain possible sources of the discrepancy.

Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.

Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.