Probability from 0 to 1 and the expected count

The probability of an event is always a number from 0 to 1. The reason: the number of times the event happened is at most the number of trials, so the fraction $\displaystyle \frac{f}{n}$ is not less than 0 and not more than 1.

• The closer the probability is to 1, the more likely the event.
• A probability of $\displaystyle \frac{1}{2}$, that is 50%: the event happens in about half of the times, like heads when tossing a coin.
• In percent: from 0% to 100%.

At the two ends of the scale there are two kinds of events:

• An impossible event never happens, and its probability is 0. For example: rolling an ordinary die gives a 7.
• A certain event happens every time, and its probability is 1, that is 100%. For example: rolling a die gives a number less than 7.
• All the other events are between 0 and 1. For example: rolling a die gives an even number – $\displaystyle {\frac{3}{6}}={\frac{1}{2}}$.

Dan rolled a die 8 times, and a 6 never came up. The relative frequency of 6 is $\displaystyle {\frac{0}{8}}={0}$, but the event “a 6 comes up” is not impossible: its probability is $\displaystyle \frac{1}{6}$, and by chance it did not happen in the eight rolls.

• A relative frequency of 0 or 1 in a few trials does not mean the event is impossible or certain.

When you know the probability of an event, you estimate how many times it will happen in many trials: multiply the probability by the number of trials.

• A die is rolled 300 times. The probability of a 6 is $\displaystyle \frac{1}{6}$, so a 6 is expected to come up about $\displaystyle {\frac{1}{6}\cdot300}={50}$ times.
• The bottle lands standing with a probability of about 0.2. In 50 flips it is expected to land standing about $\displaystyle {0.2\cdot50}={10}$ times.
• “Expected” is not “exactly”: perhaps the 6 comes up 47 times, perhaps 55 – but close to 50.

The forecast for the ten days of the holiday: “Each day the probability of rain is 30%.”
How many rainy days are expected? $\displaystyle {30\%\cdot10}={\frac{30}{100}\cdot10}={3}$: about 3 days.

For each event, write whether it is certain, impossible or neither:

• Rolling a die gives a number greater than 0.
• A marble is taken from a bag holding only blue marbles, and it is red.
• It will rain tomorrow.
• One toss of a coin gives both heads and tails.

These are the probabilities of four events:
$\displaystyle 0.35$, $\displaystyle \frac{3}{5}$, $\displaystyle 40\%$, $\displaystyle \frac{1}{4}$
Order them from the most likely event to the least likely.

A die is rolled 600 times.
About how many times will a 3 come up?

A football player scores a penalty with a probability of 0.85.
About how many penalties out of 40 will he score?

A factory expects about 12 faulty phones in every 400 phones.
What is the probability that a phone picked at random is faulty? Write it as a percentage.

A bottle lands standing with a probability of 0.2.
How many times must it be flipped to be expected to land standing about 30 times?