Probability with Coins, Dice and Cards: Quick Revision

A brief note on Probability with Coins, Dice and Cards. Read it before a test. Try each question at the end before you open its answer.

Coins and dice

• Two coins: HH, HT, TH, TT. HT and TH are different outcomes.
• Three coins: $8$ outcomes. Each extra coin doubles the count.
• Two dice: $36$ pairs (first, second). Sums are not equally likely: $7$ comes in $6$ ways, $2$ and $12$ in one way each.
• "At least one" is usually quickest by the complement.

A pack of cards

• $52$ cards, $4$ suits of $13$. Hearts and diamonds are red, spades and clubs black.
• $12$ face cards (jack, queen, king). $4$ cards of each kind. The ace is not a face card.
• In an "or" question, count a card in both groups once.
• When cards are removed, divide by the cards that are left.

Bags, discs and spinners

• 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.
• Unequal sectors on a spinner are not equally likely.

A worked example

Two dice are rolled. Find the probability that the sum is $5$ or less.

1. Count each sum. Sum $2$: $1$ way. Sum $3$: $2$. Sum $4$: $3$. Sum $5$: $4$.
2. Add. $1 + 2 + 3 + 4 = 10$ outcomes.
3. Divide. $\dfrac{10}{36} = \dfrac{5}{18}$.

Always simplify the fraction at the end.

Practice with new numbers

A program to try

The program counts outcomes for three coins and for two dice by listing them.

Mistakes to avoid

• Three outcomes for two coins in place of four.
• Treating every sum of two dice as equally likely.
• Counting a card twice in an "or" question.
• Dividing by $52$ after cards have been removed.

Check yourself

1. Two dice are rolled. What is the probability of a double?

2. One card is drawn from a full pack. What is the probability that it is a black ace?

3. A bag holds $4$ red and $6$ green balls. What is the probability of a green ball?

4. Three coins are tossed. What is the probability of at least one tail?