Probability · Ordinary Level

Two Events — And/Or

Ordinary Level · Class 4 · Tap NEXT to begin

Section 1 of 2

Both, Neither & Only One

Multiply along, add between

For two events, work out each outcome. AND = multiply the fractions (top by top, bottom by bottom); OR = add. A probability can never be over $1$.

Worked example — a spinner, twice

A spinner has $4$ equal sections $A, B, C, D$. Spin it twice. Find (i) $P(\text{two } A\text{s})$; (ii) $P(\text{an } A \text{ and a } B)$.

$4 \times 4 = 16$ possible results

(i) $A$ and $A$: $\tfrac{1}{4} \times \tfrac{1}{4} = \tfrac{1}{16}$

(ii) $AB$ or $BA$: $\tfrac{1}{16} + \tfrac{1}{16} = \tfrac{2}{16}$

$\tfrac{1}{16},\ \tfrac{2}{16}$

The probability I score a free is $\tfrac{7}{8}$ (so $P(\text{miss}) = \tfrac{1}{8}$). I take $2$ frees.

Worked example — both, neither, only the first

Find (i) $P(\text{both})$; (ii) $P(\text{neither})$; (iii) $P(\text{only the first})$.

(i) both $=$ score and score $= \tfrac{7}{8} \times \tfrac{7}{8} = \tfrac{49}{64}$

(ii) neither $=$ miss and miss $= \tfrac{1}{8} \times \tfrac{1}{8} = \tfrac{1}{64}$

(iii) only first $=$ score and miss $= \tfrac{7}{8} \times \tfrac{1}{8} = \tfrac{7}{64}$

$\tfrac{49}{64},\ \tfrac{1}{64},\ \tfrac{7}{64}$

Don’t add the tops

$\tfrac{7}{8} + \tfrac{7}{8} = \tfrac{14}{8} = \tfrac{7}{4}$ — that’s over $1$, so it’s wrong. For ‘and’ you multiply.

You try

For the two frees, find $P(\text{scores only once})$.

Score then miss, OR miss then score — work out each and add.

score and miss $= \tfrac{7}{8} \times \tfrac{1}{8} = \tfrac{7}{64}$

miss and score $= \tfrac{1}{8} \times \tfrac{7}{8} = \tfrac{7}{64}$

$\tfrac{7}{64} + \tfrac{7}{64} = \tfrac{14}{64} = \tfrac{7}{32}$

$\tfrac{7}{32}$

That’s Class 4.

Two events — both, neither and only one — using multiply for ‘and’ and add for ‘or’. Class 5: drawing with and without replacement.