Probability basics – Year 7 maths lesson to download as PDF

Example 1

A fair dice is rolled. Find the probability of a 4, an even number and a prime number.

\(P(4)=\dfrac{1}{6}\). \(P(\text{even})=\dfrac{3}{6}=\dfrac{1}{2}\) (2, 4, 6). \(P(\text{prime})=\dfrac{3}{6}=\dfrac{1}{2}\) (2, 3, 5).

Example 2

A bag has 3 red, 5 blue and 2 green counters. Find P(red), P(blue) and P(not green).

There are 10 counters. \(P(\text{red})=\dfrac{3}{10}\), \(P(\text{blue})=\dfrac{5}{10}=\dfrac{1}{2}\) and \(P(\text{not green})=\dfrac{8}{10}=\dfrac{4}{5}\).

Example 3

The probability of rain is 0.3. What is the probability of no rain?

\(1-0.3=0.7\).

Example 4

A spinner has \(P(\text{red})=0.35\) and \(P(\text{blue})=0.45\). The rest is green. Find \(P(\text{green})\).

\(1-0.35-0.45=0.2\).

Example 5

Two coins are tossed. Find the probability of one head and one tail.

The outcomes are HH, HT, TH, TT. There are 4. Two give one head and one tail, so the probability is \(\dfrac{2}{4}=\dfrac{1}{2}\).

Example 6

Two dice are rolled. Find the probability that the sum is 7.

There are 36 outcomes. The sum 7 can be made in 6 ways: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). So the probability is \(\dfrac{6}{36}=\dfrac{1}{6}\).

Example 7

A fair dice is rolled 60 times. How many sixes do you expect?

\(\dfrac{1}{6}\times60=10\) sixes.

Example 8

A drawing pin lands point up 37 times in 50 throws. Estimate the probability, and how many point-up landings you expect in 200 throws.

Relative frequency \(=\dfrac{37}{50}=0.74\). In 200 throws, \(0.74\times200=148\).

Example 9

A box holds 12 pens: 5 blue, 4 black and 3 red. Find the probability that a pen taken at random is blue. Then find the probability that the next pen is red if the blue pen is not replaced.

\(P(\text{blue})=\dfrac{5}{12}\). Black or red is the same as not blue: \(\dfrac{7}{12}\). If the blue pen is not put back, there are 11 pens left, so \(P(\text{red})=\dfrac{3}{11}\). When items are not replaced, the total changes.

Worked example in context

A bag has 3 red and 7 blue counters. What is the probability of picking a blue counter and how many blue counters would you expect in 50 picks (with replacement)?

P(blue) = \(\dfrac{7}{10}\). Expected blue: \(50\times\dfrac{7}{10}=35\).

Example 1

The letters of the word STATISTICS are written on ten cards and one card is chosen at random. Find the probability that it is the letter S, and the probability that it is a vowel.

There are ten cards and each is equally likely to be chosen. The letter S appears three times (S, S, S), so P(S) \(=\dfrac{3}{10}\). The vowels are A, I and I, which is three cards, so P(vowel) \(=\dfrac{3}{10}\) as well. As a check, the probability of a consonant is \(1-\dfrac{3}{10}=\dfrac{7}{10}\), and indeed there are seven consonants: S, T, T, S, T, C, S.

Example 2

A drawing pin is dropped 200 times and lands point up 74 times. Estimate the probability that it lands point up, and predict how many times it will land point up in 500 drops.

A drawing pin is not symmetrical, so we cannot assume the two outcomes are equally likely. Instead we use the relative frequency from the experiment: \(\dfrac{74}{200}=0.37\). This is our best estimate of the probability. For 500 drops we expect \(500\times0.37=185\) landings point up. The more times an experiment is repeated, the more reliable the relative frequency becomes.

Example 3

A fair spinner has five equal sections coloured red, red, blue, green and yellow. It is spun twice. What is the probability of getting red both times?

One way is to list the outcomes in a sample space grid with 5 rows and 5 columns, which gives 25 equally likely outcomes. Red on the first spin can happen in 2 ways and red on the second in 2 ways, so there are \(2\times2=4\) outcomes with red both times. The probability is \(\dfrac{4}{25}\). In Year 8 you will learn to multiply the probabilities directly: \(\dfrac{2}{5}\times\dfrac{2}{5}=\dfrac{4}{25}\).