# Probability · Grade 10 · 4 units

> Probability, Grade 10 · 4 units: Events, The rules of probability and more. Explanation, worked examples and free practice following the Israeli curriculum.

In Probability (Grade 10 · 4 units) you practise 8 skills: Events · The rules of probability · Venn diagrams · Conditional probability · Probability tables · Probability trees · Total probability and Bayes’ rule · Independent events.

The probability of an event is a number from zero to one. The probability of the complement is one minus the probability of the event. When events are mutually exclusive, so they cannot happen together, the probability that one of them happens is the sum of their probabilities. When they are not, subtract the probability that both happen, because it was counted twice.

A conditional probability is the probability of an event when you know another event happened: divide the probability that both happen by the probability of the known event. In a table you see it by looking only at the known event’s row or column, and in a tree it is the probability on the second branch. When you know the conditional probabilities and want the reverse one, first work out the total probability, then divide.

Two events are independent when knowing that one happened does not change the probability of the other. Check it in one of three equivalent ways: the probability of both together equals the product of the probabilities, or a conditional probability equals the plain one, in either direction.

**Tip:** Write in words what is known and what is asked before you choose a formula: the word “given” says it is a conditional probability.

## What you learn

### Events

- Example: (A square board of 100 equal cells, with a shaded rectangle) A dart lands at random on the board, with the same chance of hitting any point. What is the probability that it hits the shaded region?
- Answer: 1/10
- Solution: The shaded cells: 2 × 5 = 10 . On the whole board: 10 × 10 = 100 . The probability: 10/100 = 1/10

### The rules of probability

- Example: Each pupil in a school year chose one favourite colour. A pupil is picked at random. The probability that the pupil chose red is 0.4 and blue 0.1 . What is the probability that the pupil chose red or blue?
- Answer: 0.5
- Solution: Each pupil chose only one colour, so the events are mutually exclusive, and you add: 0.4 + 0.1 = 0.5

### Venn diagrams

- Example: A class has 25 pupils. The diagram shows how many of them play football, how many basketball, how many both and how many neither. (A Venn diagram: how many pupils play football, basketball, both or neither) A pupil is picked at random. What is the probability that the pupil plays both football and basketball? Give the answer as a fraction.
- Answer: 12/25
- Solution: The overlap of the two circles holds 12 pupils, so the probability is 12/25

### Conditional probability

- Example: The table shows the pupils in a school year by how they came to school and whether they arrived on time. (A table: pupils by how they come to school and whether they arrived on time) A pupil is picked at random. The pupil is known to have arrived on time. What is the probability that the pupil came by bus? Give the answer as a fraction.
- Answer: 1/2
- Solution: Look only at the column of the pupils who arrived on time: 5 + 5 = 10 . Of them, by bus: 5 , so the probability is 5/10 = 1/2

### Probability tables

- Example: (The table of outcomes: rows by the first spinner, columns by the second, and the sums in the cells) The first spinner has 4 equal parts, numbered 1 to 4. The second has 4 equal parts, numbered 1 to 4. Both are spun. What is the probability that the two numbers add up to 3?
- Answer: 1/8
- Solution: The number of all the outcomes: 4 × 4 = 16 . The ones you want: (1, 2) , (2, 1) : 2 outcomes. The probability: 2/16 = 1/8

### Probability trees

- Example: A bag has 6 red balls and 2 blue balls. A ball is taken out, its colour noted, and it is put back. Then a second ball is taken out. What is the probability that both balls are red?
- Answer: 9/16
- Solution: First stage: of the 8 balls, 6 are red, so the probability of a red ball is 6/8 . In the second stage the ball is back, so the probability is again 6/8 . Multiply along the branch: 6/8 × 6/8 = 36/64 = 9/16

### Total probability and Bayes’ rule

- Example: In a factory two machines make all the items. An item is picked at random. (A table: each machine’s share of the items and the chance that its item is faulty) What is the probability that the item is faulty?
- Answer: 0.034
- Solution: Add the two branches: 0.2 × 0.09 + 0.8 × 0.02 = 0.018 + 0.016 = 0.034

### Independent events

- Example: The event A and the event B are independent. Given: P(A) = 0.2 and P(B) = 0.9 . Find P(A ∩ B) .
- Answer: 0.18
- Solution: P(A ∩ B) = 0.2 × 0.9 = 0.18

## Not on the site yet

- Proving the three definitions are equivalent
- Running experiments
- Geometric probability

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 10 · 4 units topics](https://kixplo.com/il/en/math/grade-10-4u/)