Estimating probability from relative frequency
Estimating probability from relative frequency
In grade 6 you talked about chance: how likely it is that something happens. In mathematics this number is called a probability.
- An experiment is an action whose outcome is not known in advance: rolling a die, dropping a drawing pin, picking a pupil in a draw.
- An event is what you check for: “a 6 came up”, “the pin landed on its back”.
- The probability of an event is a number from 0 to 1, or from 0% to 100%, that says how likely the event is.
On a symmetric die each face has the same probability, : there is no reason for one face to come up more than another. A drawing pin, a plastic cup or a bottle has no equal sides, and the probability cannot be worked out in advance.
So it is estimated: repeat the experiment many times, and work out the event’s relative frequency.
- – the event, and – its probability.
- – the number of times the event happened.
- – the number of times the experiment was done.
In the example below we estimate the probability that a water bottle, flipped in the air, lands standing up.
Noam flipped the bottle many times, and every so often wrote down how many times it had landed standing:
| Flips | 10 | 50 | 100 | 500 | 1,000 |
|---|---|---|---|---|---|
| Landed standing | 4 | 8 | 23 | 96 | 201 |
| Relative frequency | 0.4 | 0.16 | 0.23 | 0.192 | 0.201 |
- After 10 flips the relative frequency was 0.4, and after 50 only 0.16: in a few flips luck changes a lot.
- From 100 flips on it changes little, and stays close to 0.2.
- So the probability that the bottle lands standing is about 0.2, that is about 20%.
This is how it looks when a computer drops a plastic cup 1,000 times, and checks each time whether the cup landed on its side:
- The line in the graph is the relative frequency of “on its side” after each number of drops.
- At first it jumps up and down, and later it hardly moves: the relative frequency settles.
- The reason: after many drops, one more drop changes the fraction only a little.
A relative frequency is an estimate of the probability, not the probability itself.
- If Noam flips the bottle another 1,000 times, the relative frequency will probably come out a little different, say 0.196.
- The more trials, the better the estimate: an estimate from 1,000 flips is better than one from 10 flips.
Estimate the probability that the plastic cup lands upside down.
Drop it 10 times and write down the relative frequency, then another 100 times, and then another 1,000 times.
Which estimate is the best? Why?
A basketball player scored 72 free throws out of 90 this season.
Estimate the probability that she scores her next free throw.
Write the probability as a decimal.
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
Write it as a percentage.
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
Write it as a fraction in its lowest terms.
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
Yoav dropped a drawing pin 20 times, and it landed on its back 14 times.
Michal dropped a pin of the same kind 500 times, and it landed on its back 290 times.
What is Yoav’s estimate of the probability that the pin lands on its back? Write it as a decimal.
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
What is Michal’s estimate? Write it as a decimal.
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
Whose estimate is better? Explain.
A spinner has three coloured parts, not of equal size. It was spun 400 times:
| Colour | red | blue | yellow |
|---|---|---|---|
| Frequency | 150 | 170 | 80 |
Estimate the probability that the arrow stops on yellow. Write it as a percentage.
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
Estimate the probability that the arrow stops on red or on blue. Write it as a decimal.
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power