Introduction to Probability Flashcards

All 15 cards in this deck

What is the probability of an event when all outcomes are equally likely?

$P(\text{event}) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$

e.g. one red ball out of 5 balls gives $\frac{1}{5}$

What are the smallest and largest values a probability can take, and what do they mean?

$0$ to $1$ inclusive: $0$ means impossible, $1$ means certain.

$0.5$ means an even chance.

How do you find the probability that an event does not happen?

e.g. $P(\text{rain}) = 0.3$

$P(\text{not }A) = 1 - P(A)$, so $1 - 0.3 = 0.7$

What are the steps to find a missing probability in a table of mutually exclusive, exhaustive outcomes?

e.g. $P(\text{red}) = 0.2$, $P(\text{blue}) = 0.5$, $P(\text{green}) = ?$

1. Add the known probabilities: $0.2 + 0.5 = 0.7$

2. Subtract from $1$: $1 - 0.7 = 0.3$

3. $P(\text{green}) = 0.3$

What are the steps to find a probability when the numbers of items are given as multiplicative relationships?

e.g. a bag holds only blue and yellow cubes, with twice as many blue as yellow — find $P(\text{yellow})$

1. Let yellow $= y$, so blue $= 2y$

2. Total $= y + 2y = 3y$

3. $P(\text{yellow}) = \dfrac{y}{3y} = \dfrac{1}{3}$

What are the steps to construct a sample space diagram for the total of two spins?

e.g. two fair 3-sided spinners numbered 1, 2, 3

1. Write the outcomes of one spinner along the top and the other down the side: $1, 2, 3$ each way

2. Fill each cell with the total, e.g. $2$ and $3$ gives $5$

3. The grid has $3 \times 3 = 9$ equally likely outcomes

How do you use a sample space diagram to find a theoretical probability?

e.g. $P(\text{total} = 12)$ when two ordinary dice are rolled

$\dfrac{\text{number of cells giving that outcome}}{\text{total number of cells}}$, so $\dfrac{1}{36}$

How many equally likely outcomes are there when two experiments are combined?

e.g. rolling an ordinary dice and flipping a coin

Multiply the numbers of outcomes: $6 \times 2 = 12$

True or false? When two ordinary dice are rolled, every total from 2 to 12 is equally likely.

False. The 36 outcomes are equally likely, but six of them give a total of 7 and only one gives 2.

True or false? In the sample space for rolling two dice, $(2,5)$ and $(5,2)$ count as the same outcome.

False. They are two different outcomes and must both be counted.

What is the formula for relative frequency (an estimate of probability from experimental data)?

$\dfrac{\text{frequency of the event}}{\text{total number of trials}}$

e.g. 15 sixes in 60 throws gives $\frac{15}{60} = 0.25$

How do you work out an expected frequency?

e.g. $P(\text{A}) = 0.4$ and the spinner is spun 200 times

Expected frequency $=$ probability $\times$ number of trials, so $0.4 \times 200 = 80$

What happens to the relative frequency of an outcome as the number of trials increases, for an unbiased experiment?

It gets closer to the theoretical probability.

True or false? A dice that lands on a six 3 times in 10 throws must be biased.

False. With so few trials results vary by chance; you need many more trials before judging bias.

True or false? If you calculate an expected frequency of 50 sixes in 300 rolls, the dice may still land on six a different number of times.

True. Expected frequency is only an estimate of what to expect on average, not a guarantee of the actual result.