Probability: outcomes and sample spaces – Year 8 maths lesson to download as PDF

Worked example 1: a die and a bag

Find \(P(5)\) and \(P(\text{even})\) for a fair die. A bag has 3 red, 5 blue and 2 green counters; find \(P(\text{blue})\) and \(P(\text{red or green})\).

\(P(5)=\dfrac16\) and \(P(\text{even})=\dfrac36=\dfrac12\). The bag holds 10 counters, so \(P(\text{blue})=\dfrac5{10}=\dfrac12\) and \(P(\text{red or green})=\dfrac5{10}=\dfrac12\).

Worked example 2: the missing outcome

\(P(A)=0.35\) and \(P(B)=0.4\) are two of three possible outcomes. Find the third probability, and \(P(\text{not rolling a }6)\).

The third is \(1-0.35-0.4=0.25\). The probability of not rolling a 6 is \(1-\dfrac16=\dfrac56\).

Worked example 3: is the die fair?

A die is rolled 120 times and shows a 6 on 28 occasions. Is it fair?

Relative frequency \(=\dfrac{28}{120}\approx0.233\). A fair die would give \(120\times\dfrac16=20\) sixes, so the die may be biased, but 120 rolls is not many, so you would want more trials.

Worked example 4: expected wins

A spinner wins with probability 0.2. How many wins do you expect in 150 spins? A coin gives 63 heads in 100 tosses; how many heads in 500 tosses?

\(0.2\times150=30\) wins. The estimate for heads is 0.63, so in 500 tosses expect about \(0.63\times500=315\) heads.

Worked example 5: coins

List the outcomes of two coins and find \(P(\text{exactly one head})\). Then find \(P(\text{at least one head})\) with three coins.

Two coins give HH, HT, TH, TT, four outcomes, so \(P(\text{exactly one head})=\dfrac24=\dfrac12\). Three coins give 8 outcomes, so \(P(\text{at least one head})=1-P(\text{TTT})=1-\dfrac18=\dfrac78\).

Worked example 6: two dice

Two fair dice are rolled and the scores are added. Find \(P(\text{sum}=7)\) and \(P(\text{sum}\ge10)\).

Draw a 6 by 6 grid with one die along the top and the other down the side, filling in the totals. There are 36 equally likely outcomes. A total of 7 appears 6 times, so \(P(\text{sum}=7)=\dfrac6{36}=\dfrac16\). A total of at least 10 comes from (4,6), (5,5), (6,4), (5,6), (6,5) and (6,6), which is also \(\dfrac6{36}=\dfrac16\).

Worked example 7: Venn diagram

In a class of 30, 18 play football, 12 play tennis and 5 play both. Find \(P(\text{neither})\) and \(P(\text{football or tennis})\).

Start with the overlap: 5 play both. Football only is \(18-5=13\) and tennis only is \(12-5=7\). Together these cover \(13+5+7=25\), so 5 play neither. \(P(\text{neither})=\dfrac5{30}=\dfrac16\) and \(P(\text{football or tennis})=\dfrac{25}{30}=\dfrac56\).

Worked example 8: sets of numbers

For the numbers 1 to 12, A is the even numbers and B the multiples of 3. Find \(A\cap B\), \(A\cup B\) and \(P(A\cup B)\).

\(A\cap B=\{6,12\}\) and \(A\cup B=\{2,3,4,6,8,9,10,12\}\), so \(P(A\cup B)=\dfrac8{12}=\dfrac23\).

Worked example 9: probabilities that add up to 1

A spinner can land on red, blue or green. \(P(\text{red})=0.3\) and \(P(\text{blue})=0.45\). Find \(P(\text{green})\) and the expected number of greens in 80 spins.

The probabilities add up to 1, so \(P(\text{green})=1-0.3-0.45=0.25\). Expected greens \(=0.25\times80=20\).

Worked example 10: a sample space table

Two spinners, each with 1, 2, 3, are spun and the scores added. Find the probability that the total is 4.

List all \(3\times3=9\) outcomes: totals 2, 3, 4 / 3, 4, 5 / 4, 5, 6. Total 4 occurs three times (1+3, 2+2, 3+1). So \(P(4)=\dfrac{3}{9}=\dfrac13\).

Worked example 11: a two-circle Venn diagram

In a class of 28, 15 like pizza (P), 12 like pasta (Q) and 5 like both. How many like neither?

Both \(=5\). Pizza only \(=15-5=10\). Pasta only \(=12-5=7\). At least one \(=10+5+7=22\). Neither \(=28-22=6\). \(P(\text{neither})=\dfrac{6}{28}=\dfrac{3}{14}\).

Worked example 12: choosing from relative frequency

A bag has red and blue counters. Sam picks a counter 100 times (replacing it each time) and gets red 38 times. Estimate the number of red counters if there are 50 counters in total.

Relative frequency of red \(=\dfrac{38}{100}=0.38\). Expected reds \(=0.38\times50=19\). This is an estimate: the true number might be 18 or 20.

Example: theory or experiment?

A drawing pin lands point up 62 times in 200 drops. A fair coin lands heads 104 times in 200 flips. Which probabilities should we use?

For the coin we can calculate a theoretical probability, \(\dfrac{1}{2}\), because heads and tails are equally likely; the 104 heads (0.52) is just normal variation. For the drawing pin the two outcomes are not equally likely, so there is no theoretical value and the relative frequency \(\dfrac{62}{200}=0.31\) is our best estimate. Repeating the experiment more times would make this estimate more reliable.