Introduction to Probability Flashcards

All 23 cards in this deck

What is the range of values a probability can take?

From $0$ to $1$ inclusive.
$0$ = impossible, $1$ = certain.

Which word describes a probability of exactly $\frac{1}{2}$ on the probability scale?

Evens (even chance).

How do you write the probability of an event when all outcomes are equally likely?

e.g. 4 red sweets out of 15 sweets

$P(\text{event}) = \dfrac{\text{number of successful outcomes}}{\text{total number of outcomes}}$
e.g. $\frac{4}{15}$

True or false? A probability may be written as a ratio such as $4:15$.

False.
Probability must be a fraction, decimal or percentage — $4:15$ scores no marks.

What is the formula for the probability of an event $A$ not happening?

$P(\text{not }A) = 1 - P(A)$
e.g. if $P(\text{six}) = 0.3$ then $P(\text{not six}) = 0.7$

What do the probabilities of a set of mutually exclusive outcomes that covers everything (exhaustive) add up to?

$1$.
So a missing probability $= 1 -$ (the others).

What does it mean for two events to be mutually exclusive, and what can you then do with their probabilities?

They cannot both happen at the same time; then $P(A\text{ or }B) = P(A)+P(B)$.

True or false? For a single roll of a dice, 'even number' and 'greater than 3' are mutually exclusive, so you can add their probabilities.

False.
4 and 6 are both, so the events overlap and adding is invalid.

What are the steps to find the total number of items from a given probability?

e.g. $P(\text{blue}) = 0.2$ and there are 12 blue cubes

1. Write probability $=\dfrac{\text{number of that type}}{\text{total}}$: $0.2 = \frac{12}{\text{total}}$
2. Divide the number by the probability: $12 \div 0.2$
3. Total $= 60$ cubes

What does 'taken at random' mean in a probability question?

Every item is equally likely to be chosen — no item is favoured.

What is a sample space (possibility space) diagram?

A list or grid showing all the possible outcomes of an experiment.

How do you list all possible outcomes systematically?

e.g. a coin thrown 3 times

Work in a fixed order, changing the last item first: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

How many outcomes are there in total when two ordinary six-sided dice are rolled?

$36$ ($6 \times 6$).

What are the steps to find a probability using a sample space grid?

e.g. $P(\text{total} = 4)$ when two four-sided spinners are spun

1. Draw a grid with one experiment along the top, the other down the side: $4 \times 4$ grid
2. Fill every cell with the combined outcome (here the total) — $16$ cells
3. $P = \dfrac{\text{cells wanted}}{\text{total cells}} = \frac{3}{16}$

True or false? In a sample space diagram for two dice, the outcomes (2, 5) and (5, 2) are the same outcome and should be counted once.

False.
They are separate cells in the grid and must both be counted.

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

False.
The 36 outcomes are equally likely, but some totals come from more cells than others.

What is relative frequency (experimental probability)?

$\dfrac{\text{number of times the outcome happened}}{\text{total number of trials}}$

What is the formula for expected frequency (an estimate of the number of successes)?

Expected frequency $=$ probability $\times$ number of trials.
e.g. $0.2 \times 60 = 12$

Whose results give the best estimate of a probability, and why?

The person with the greatest number of trials — a larger sample gives a better estimate.

True or false? An expected frequency tells you exactly how many times the outcome will occur.

False.
It is only an estimate of the number of times.

True or false? As the number of trials increases, the relative frequency tends to get closer to the theoretical probability.

True.
Larger unbiased samples tend towards the theoretical probability.

What are the steps to decide whether a dice is fair or biased from experimental results?

e.g. a dice is rolled 60 times and lands on six 25 times

1. Theoretical probability if fair: $\frac{1}{6}$
2. Expected frequency: $\frac{1}{6}\times 60 = 10$
3. Compare with the relative frequency/actual result: 25 is much greater than 10, so the dice is probably biased.

True or false? A spinner with red, blue, green and yellow sections that lands on red 7 times out of 20 spins must be biased.

False. A small difference from the expected result can happen by chance; bias is only suggested when the relative frequency is very different from the theoretical probability over many trials.