Bags, discs and spinners

Balls in a bag

A bag holds $5$ red, $3$ blue and $4$ green balls, all the same size. Arjun draws one without looking, so each of the $12$ balls is equally likely.

• $P(\text{red}) = \dfrac{5}{12}$.
• $P(\text{blue}) = \dfrac{3}{12} = \dfrac{1}{4}$.
• $P(\text{not green}) = \dfrac{5 + 3}{12} = \dfrac{2}{3}$, or $1 - \dfrac{4}{12}$ by the complement.

Working back from a probability

A bag holds $6$ white balls and some black balls. The probability of a black ball is $\dfrac{2}{3}$. How many black balls are there?

1. Name the unknown. Let there be $x$ black balls, so $6 + x$ balls in all.
2. Write the equation. $\dfrac{x}{6 + x} = \dfrac{2}{3}$.
3. Solve. $3x = 12 + 2x$, so $x = 12$.
4. Check. $\dfrac{12}{18} = \dfrac{2}{3}$.

Adding balls

A bag holds $4$ red and $8$ blue balls. How many red balls must be added so that a red ball and a blue ball are equally likely?

The two colours must then be equal in number, so the bag needs $8$ red balls. Add $4$ red balls. Now $P(\text{red}) = \dfrac{8}{16} = \dfrac{1}{2}$.

Numbered discs

A box holds discs numbered $1$ to $50$. Meera picks one at random.

• A multiple of $7$: $7, 14, \ldots, 49$ is $7$ discs, so $P = \dfrac{7}{50}$.
• A two-digit number: $10$ to $50$ is $41$ discs, so $P = \dfrac{41}{50}$.
• A number above $50$: none, so $P = 0$. That is an impossible event.

Spinners

A spinner has $8$ equal sectors numbered $1$ to $8$. The arrow is equally likely to stop on each.

• $P(\text{an odd number}) = \dfrac{4}{8} = \dfrac{1}{2}$.
• $P(\text{a number greater than } 6) = \dfrac{2}{8} = \dfrac{1}{4}$.

Equal sectors matter. If one sector is twice as wide as the others, the arrow stops there more often. Then the outcomes are not equally likely, and the simple rule does not apply.

A calendar question

A year that is not a leap year has $365$ days. That is $52$ weeks and $1$ extra day. The extra day is equally likely to be any of the $7$ days of the week. So the probability that such a year has $53$ Sundays is $\dfrac{1}{7}$.

Practise with new numbers

Try it in Python

The program draws a ball from the bag many times, with replacement, and compares the share of red with $\dfrac{5}{12}$.

Common mistakes

• Forgetting the balls already in the bag. The total is all the balls, not just one colour.
• Unequal sectors. A spinner with sectors of different sizes does not give equally likely outcomes.
• Miscounting the two-digit numbers. From $10$ to $50$ there are $41$ numbers, not $40$.
• A probability above $1$. If your answer is bigger than $1$, the count is wrong.

Remember these

• The total is every ball, disc or sector that could come up.
• From a probability, write an equation in the unknown number and solve it.
• Check that the outcomes are equally likely before you divide.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A bag holds $3$ yellow and $9$ purple balls. What is the probability of a yellow ball?

2. Discs numbered $1$ to $50$ are in a box. What is the probability of a perfect square?

3. A bag holds $10$ balls, and the probability of a green one is $\dfrac{2}{5}$. How many green balls are there?

4. A spinner has $6$ equal sectors numbered $1$ to $6$. What is the probability that it stops on a number less than $3$?

5. Can a bag have $P(\text{red}) = \dfrac{3}{8}$ and $P(\text{not red}) = \dfrac{1}{2}$?