Two events together

Two dice

Roll two dice and add the numbers. What is the chance of a sum of $7$? List all the pairs: the first die can be $1$ to $6$, and so can the second. That makes $6 \times 6 = 36$ pairs, all equally likely.

Six of the $36$ pairs make $7$, so $P(\text{sum} = 7) = \dfrac{6}{36} = \dfrac{1}{6}$. A sum of $7$ is the most likely. A sum of $2$ has only one pair, $(1, 1)$.

Try it in Python

Two coins

Toss two coins. The outcomes are HH, HT, TH and TT, all four equally likely. So $P(\text{two heads}) = \dfrac{1}{4}$, and $P(\text{at least one head}) = \dfrac{3}{4}$.

For events that do not affect each other, there is a short cut: the chance of both is the product of the two chances. For two heads, $0.5 \times 0.5 = 0.25$.

At least one

"At least one six in three rolls" is hard to count directly, because there are many ways. It is easy the other way round: find the chance of no six, and take it from $1$.

You should find that the curve rises fast at first and then flattens, always below 1. With a chance of 0.17 a try, you need about 4 tries for an even chance of at least one success.

The program checks the answer for three rolls with a simulation.

Taking two marbles from the bag

The bag has $3$ red marbles among $10$. You take two.

• If you put the first back, the chances do not change: $P(\text{both red}) = \dfrac{3}{10} \times \dfrac{3}{10} = 0.09$.
• If you do not put it back, the bag changes. After a red one, $2$ red marbles are left among $9$: $P(\text{both red}) = \dfrac{3}{10} \times \dfrac{2}{9} = \dfrac{1}{15} \approx 0.067$.

Practise with new numbers

Common mistakes

• Listing 11 sums of two dice as equally likely. The 36 pairs are equally likely; the sums are not.
• Adding chances for "both". For both of two separate events, multiply.
• Counting "at least one" the long way. Take the chance of none from 1.
• Forgetting that the bag changes. Without putting back, the second chance is different.

Remember these

• List all the equally likely outcomes, then count the ones that you want.
• Two dice have $36$ pairs. A sum of $7$ has the most pairs.
• At least one: $1$ minus the chance of none.
• Without putting back, the second chance changes.

Check yourself

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

1. Two coins are tossed. What is the probability of two tails?

2. What is the probability that two dice add up to $12$?

3. A coin is tossed 3 times. What is the probability of at least one head?

4. Two cards are drawn from 5 cards numbered 1 to 5, without putting the first back. What is the probability that both are even?