What is the difference between an outcome and an event?

An outcome is a single possible result of a trial, such as a dice landing on 6. An event is a collection of outcomes, so "the dice lands on an even number" is one event made up of the three outcomes 2, 4 and 6.

True or False? Counting favourable outcomes over total outcomes gives the probability for any experiment.

False. It only works when the outcomes are equally likely , which is what makes an experiment fair. If some outcomes are more likely than others the experiment is biased , and counting alone will not give the right answer.

Items are drawn from a bag and not replaced. What two things change before the next draw?

The total number of items goes down by one for every item already drawn. The number of items satisfying the event also goes down, but only by however many of the drawn items actually satisfied it. Both the top and the bottom of the fraction have to be recounted, and forgetting the top is the usual slip.

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

Work out the probability that it does happen and subtract that from 1. Alternatively, count the outcomes that do not cause the event and write that over the total number of outcomes, which gives the same answer.

Define sample space diagram .

A sample space diagram shows all the possible outcomes of an experiment. It can be written as a list or set out as a table, and it has to be complete for any probability read from it to be right.

When is a two-way table the right kind of sample space diagram?

When two experiments happen together, such as rolling two dice or picking a day of the week and rolling a dice. One experiment's outcomes go along the top and the other's down the side, and each cell holds the combined result.

Once a sample space table is filled in, how do you find the outcomes for the event?

Go through every entry and mark the ones that satisfy the event, by circling, underlining or ticking them. Then count the marks: that count is the numerator, and the total number of entries is the denominator.

Define relative frequency .

Relative frequency is an estimate of a probability worked out from the results of an experiment. It is the number of times the event actually happened divided by the total number of trials.

True or False? Relative frequency gets closer to the actual probability as more trials are carried out.

True. A small experiment can be unrepresentative purely by chance, so its estimate may be well out. As the number of trials grows the estimate settles down and closes in on the actual probability.

How do you decide whether an actual result is more or less than expected?

Either work out the expected frequency and compare it with the number that actually occurred. Or work out the relative frequency and compare it with the probability you were given. The two routes are equivalent, so use whichever the numbers make easier.

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Probability (SQA National 5 Applications of Mathematics): Flashcards

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What is the difference between an outcome and an event?

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Cards in this collection (20)

What is the difference between an outcome and an event?
An outcome is a single possible result of a trial, such as a dice landing on 6.
An event is a collection of outcomes, so "the dice lands on an even number" is one event made up of the three outcomes 2, 4 and 6.
Complete the probability of an event when the outcomes are equally likely:
$P = \frac{number of outcomes for the______}{total number of______}$
The completed fraction is:
$P = \frac{number of outcomes for the event}{total number of outcomes}$
Both counts have to come from the same list of possibilities, which is why the total is counted first.
A whole number is picked at random from 1 to 49. What is the probability that it is a multiple of 5?
The probability is $\frac{9}{49}$ .
There are 49 possible outcomes altogether, and 9 of them are multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40 and 45.
True or False?
Counting favourable outcomes over total outcomes gives the probability for any experiment.
False.
It only works when the outcomes are equally likely, which is what makes an experiment fair.
If some outcomes are more likely than others the experiment is biased, and counting alone will not give the right answer.
Items are drawn from a bag and not replaced. What two things change before the next draw?
The total number of items goes down by one for every item already drawn.
The number of items satisfying the event also goes down, but only by however many of the drawn items actually satisfied it.
Both the top and the bottom of the fraction have to be recounted, and forgetting the top is the usual slip.
A bag holds balls numbered 1 to 30, of which 9 end in 3, 6 or 9, and the first four drawn without replacement are 4, 18, 16 and 29. What is the probability the fifth ends in 3, 6 or 9?
The probability is $\frac{7}{26}$ .
Four balls have gone, so $30 - 4 = 26$ remain.
Of the four drawn, only 16 and 29 were scoring balls, so $9 - 2 = 7$ of them are left.
How do you find the probability that an event does not happen?
Work out the probability that it does happen and subtract that from 1.
Alternatively, count the outcomes that do not cause the event and write that over the total number of outcomes, which gives the same answer.
Define sample space diagram.
A sample space diagram shows all the possible outcomes of an experiment.
It can be written as a list or set out as a table, and it has to be complete for any probability read from it to be right.
When is a two-way table the right kind of sample space diagram?
When two experiments happen together, such as rolling two dice or picking a day of the week and rolling a dice.
One experiment's outcomes go along the top and the other's down the side, and each cell holds the combined result.
Two spinners have 5 and 6 sectors. How many outcomes are there altogether?
There are 30 outcomes.
Every one of the 5 sectors on the first spinner can pair with every one of the 6 on the second, so the two numbers are multiplied: $5 \times 6 = 30$ .
True or False?
A sample space table for the total on two dice has 36 entries, even though the totals only run from 2 to 12.
True.
The entries are the equally likely outcomes, and there are $6 \times 6 = 36$ of them.
Several different pairs give the same total, so the eleven possible totals are not equally likely and must never be used as the denominator.
Two spinners showing 0, 1, 2, 4, 8 and 0, 1, 2, 4, 5, 10 are spun and the numbers multiplied. What is the probability the product is greater than 10?
The probability is $\frac{4}{15}$ .
The table has $5 \times 6 = 30$ entries, and 8 of the products come to more than 10.
That gives $\frac{8}{30}$ , which simplifies to $\frac{4}{15}$ .
Once a sample space table is filled in, how do you find the outcomes for the event?
Go through every entry and mark the ones that satisfy the event, by circling, underlining or ticking them.
Then count the marks: that count is the numerator, and the total number of entries is the denominator.
Complete the rule for expected frequency:
Expected frequency is found by multiplying the $______$ by the number of $______$ carried out.
The completed rule is:
Expected frequency is found by multiplying the probability by the number of trials carried out.
So flipping a fair coin 100 times gives an expected $0.5 \times 100 = 50$ heads.
The probability of winning a mini-game is 0.65 and it is played 60 times. How many wins would you expect?
You would expect 39 wins.
Multiplying the probability by the number of trials gives $0.65 \times 60 = 39$ .
An expected frequency is a prediction, not a guarantee: the actual number of wins will often differ from it.
Define relative frequency.
Relative frequency is an estimate of a probability worked out from the results of an experiment.
It is the number of times the event actually happened divided by the total number of trials.
True or False?
Relative frequency gets closer to the actual probability as more trials are carried out.
True.
A small experiment can be unrepresentative purely by chance, so its estimate may be well out.
As the number of trials grows the estimate settles down and closes in on the actual probability.
An unfair coin is flipped 50 times and lands on heads 20 times. What can you say about the probability of heads?
The best estimate is $\frac{20}{50}$ , which is 0.4.
Because the coin is unfair you cannot work the probability out by counting equally likely outcomes, so the experiment is all you have.
It remains an estimate: the actual probability is not known, and more flips would give a better one.
How do you decide whether an actual result is more or less than expected?
Either work out the expected frequency and compare it with the number that actually occurred.
Or work out the relative frequency and compare it with the probability you were given.
The two routes are equivalent, so use whichever the numbers make easier.
A player wins 3 stars per win, the probability of winning is 0.65, and 60 games give 114 stars. Is that more or less than expected?
It is fewer stars than expected.
The expected number of wins is $0.65 \times 60 = 39$ , which at 3 stars each comes to $39 \times 3 = 117$ stars.
Since 114 is below 117, the player did slightly worse than expected.

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