Sec 2 · G1 (N(T))

Probability

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1. Probability of single events

Term	Meaning	Example: a fair die
sample space	all the possible outcomes	$\{1, 2, 3, 4, 5, 6\}$
probability	$P(E) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$	$P(\text{even}) = \dfrac{3}{6}$ , so $\dfrac{1}{2}$
impossible	$P(E) = 0$	$P(7) = 0$
certain	$P(E) = 1$	$P(\text{less than } 7) = 1$
not happening	$P(\text{not } E) = 1 - P(E)$	$P(\text{not } 6) = \dfrac{5}{6}$

A card from 1 to 20: P(a multiple of 3)

\displaystyle P = \dfrac{6}{20}

e.g.

\displaystyle \dfrac{6}{20}

\displaystyle = \dfrac{3}{10}

A bag of 4 red and some blue: P(red) is one third

\displaystyle \dfrac{4}{4 + b} = \dfrac{1}{3}

e.g.

\displaystyle \dfrac{4}{4 + b} = \dfrac{1}{3}

\displaystyle 12 = 4 + b

\displaystyle b = 8

📖 Worked example

Cards numbered $1$ to $15$ are shuffled and one is drawn at random. Find the probability that the number is a prime number.

Step 1 of 2

List the numbers that are a prime number

2, 3, 5, 7, 11, 13

That is

6

numbers out of

15

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1. Probability of single events

Term	Meaning	Example: a fair die
sample space	all the possible outcomes	$\{1, 2, 3, 4, 5, 6\}$
probability	$P(E) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$	$P(\text{even}) = \dfrac{3}{6}$ , so $\dfrac{1}{2}$
impossible	$P(E) = 0$	$P(7) = 0$
certain	$P(E) = 1$	$P(\text{less than } 7) = 1$
not happening	$P(\text{not } E) = 1 - P(E)$	$P(\text{not } 6) = \dfrac{5}{6}$

A card from 1 to 20: P(a multiple of 3)

\displaystyle P = \dfrac{6}{20}

e.g.

\displaystyle \dfrac{6}{20}

\displaystyle = \dfrac{3}{10}

A bag of 4 red and some blue: P(red) is one third

\displaystyle \dfrac{4}{4 + b} = \dfrac{1}{3}

e.g.

\displaystyle \dfrac{4}{4 + b} = \dfrac{1}{3}

\displaystyle 12 = 4 + b

\displaystyle b = 8

📖 Worked example

Cards numbered $1$ to $15$ are shuffled and one is drawn at random. Find the probability that the number is a prime number.

Step 1 of 2

List the numbers that are a prime number

2, 3, 5, 7, 11, 13

That is

6

numbers out of

15

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