Chance from data

What happened in a larger part of the times before is more likely to happen next time too. That part is measured by the relative frequency.

• In class 6B, 9 pupils out of 24 chose a dog, and 2 out of 24 a parrot.
• Ask a pupil from another class at the same school: it is more likely that they choose a dog than a parrot.
• It is an estimate, not a certainty: the pupil may choose a parrot too.

The chance that something happens is a number from 0 to 1 – or from 0% to 100%. The larger the number, the more likely it is to happen.

• 0 – impossible: it cannot happen. Next week will have eight days.
• Near 0 – unlikely: it can happen, but only rarely. It snows in Tel Aviv in summer.
• $\displaystyle \frac{1}{2}$ – “fifty-fifty”: the same chance that it happens and that it does not.
• Near 1 – likely: it happens most of the time. It rains in the Galilee in January.
• 1 – certain: it will surely happen. Monday comes after Sunday.

The example below compares the chance of rain on the day of a trip, if the trip is in January or in February.
10 rainy days are expected in January, and 9 rainy days in February.
January has 31 days, and February has 28 days.

• The chance of rain on a day in January is about 10 out of 31: $\displaystyle \frac{10}{31}\approx0.32$ (the sign $\displaystyle \approx$ means “about”).
• The chance of rain on a day in February is about 9 out of 28: $\displaystyle \frac{9}{28}\approx0.32$.
• January has more rainy days, but also more days in the month – and the chance is almost the same: about 32%, a little less than a third.
• In both months it is more likely that it does not rain on the day of the trip.

The example below chooses a computer for the show at the end-of-year party: the one with the smallest chance of a fault.
The class has three computers.
During the year the class wrote down on how many days each computer was used, and on how many of those days it had a fault:

• Computer A: $\displaystyle {\frac{6}{200}}={\frac{3}{100}}={3\%}$.
• Computer B: $\displaystyle {\frac{4}{40}}={\frac{10}{100}}={10\%}$.
• Computer C: $\displaystyle {\frac{5}{100}}={5\%}$.
• Computer B had the fewest faults, but it was also used the least. Its chance of a fault is the largest.
• The chance of a fault is smallest in computer A – the one to use for the show.

The school basketball team won 12 of its 20 games this year.
By the data, what is the chance that the team wins its next game?

Write it as a fraction in its lowest terms.

Write it as a percentage.

Which is more likely – that the team wins its next game, or that it does not? Explain.

Dana scored 12 free throws out of 20.
Noa scored 15 free throws out of 30.
Which of them is more likely to score her next free throw? Explain.

Yoav can go to school on bus 5 or on bus 12.
Last month bus 5 was late on 6 days out of 20.
Bus 12 was late on 8 days out of 40.
Which bus should Yoav take, so that his chance of being late is smaller? Explain.

For each sentence, write whether it is impossible, unlikely, “fifty-fifty”, likely or certain.

• A tossed coin lands heads.
• It is hot at the Dead Sea in August.
• Tomorrow you meet an elephant on your way to school.
• A year has 12 months.
• All 30 pupils in the class have their birthday on the same day.

The weather forecast said: “Tomorrow there is a 70% chance of rain.”
What is the chance that it does not rain tomorrow?