Experiments: a coin and a die

An experiment is an action whose result you cannot know in advance: tossing a coin, rolling a die, spinning a dreidel.
Each thing that can come out of the experiment is an outcome.

• Tossing a coin has two outcomes: heads and tails.
• Rolling a die has six outcomes: 1, 2, 3, 4, 5, 6.

A coin is symmetric: its two sides have the same shape and the same weight, and there is no reason for it to land on one of them rather than the other.
So the chance of heads equals the chance of tails – “fifty-fifty”: $\displaystyle \frac{1}{2}$ each.
Try it yourself:

• Toss the coin 10 times: you will not necessarily get exactly 5 heads and 5 tails.
• Toss it 1,000 more times: heads comes up in about half of the tosses, and tails in the other half.

The chance is a half only when the two possibilities are equally likely – like the two sides of a coin. In many cases with two possibilities the chances differ:

• A drawing pin dropped on the table lands on its back (point up) or on its side. A pin is not symmetric, so there is no reason for the chances to be equal.
• A shot at the basket goes in or does not. A good player has a chance larger than a half of scoring.
• It snows in Tel Aviv tomorrow – or it does not. The chance of snow is close to 0.

A die is symmetric too: its six faces are the same size, and there is no face it tends to land on. So each of the six outcomes has the same chance: $\displaystyle \frac{1}{6}$.

• The chance of a 1: $\displaystyle \frac{1}{6}$.
• The chance of a 4: $\displaystyle \frac{1}{6}$ too. The number written on a face does not change its chance.

An event is what you ask about in an experiment, and it may hold several outcomes.
The event “an even number comes up” happens in three outcomes: 2, 4 and 6.

When all the outcomes are equally likely, the chance of an event is:

• $\displaystyle k$ – the number of outcomes in which the event happens.
• $\displaystyle n$ – the number of all the outcomes.
• The chance of an even number: $\displaystyle {\frac{3}{6}}={\frac{1}{2}}$ – “fifty-fifty”, like heads on a coin.

Roll the die, and see in how many of the rolls an even number came up:

• The line in the graph is the relative frequency of an even number after each number of rolls.
• At first it jumps up and down. After many rolls it stays close to $\displaystyle \frac{1}{2}$.

A die is rolled.
What is the chance of each event?
Write each chance as a fraction in its lowest terms.

A 5 comes up.

A number larger than 4 comes up.

An odd number comes up.

Which event is impossible, and which is certain: “a 7 comes up”, “a number smaller than 7 comes up”? What is the chance of each?

A spinner is divided into four equal parts: two red, one blue and one yellow.
The arrow is spun, and you look at the colour it stops on.

What is the chance that the arrow stops on red? Write it as a fraction in its lowest terms.

Tal says: “There are three colours, so the chance of blue is $\displaystyle \frac{1}{3}$.”
Is Tal right? Explain.
Spin the spinner many times and check.

A bag holds 3 red marbles and one blue marble, all the same size.
One marble is taken out without looking.
What is the chance that the marble is blue?

Omer bought a raffle ticket.
He says: “Either I win or I don’t – I have a fifty-fifty chance.”
What is Omer’s mistake?

Roll the die 10 times.
Did each of the numbers come up at least once?
Roll it 1,000 more times, so that there are 1,010 rolls.
About how many times did each number come up?
Explain why.