Dice Sums, Matching and Geometric Probability: Quick Revision

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

Dice and repeated trials

• Two dice: a total of $s$ has $6 - |s - 7|$ ways out of $36$.
• Three dice: the ways for the totals $3$ to $18$ are $1, 3, 6, 10,$ $15, 21, 25, 27,$ $27, 25, 21, 15,$ $10, 6, 3, 1$ out of $216$.
• Exactly $k$ heads in $n$ tosses: $\dfrac{^nC_k}{2^n}$.
• At least one in $n$ trials: $1 - (\text{chance of none in one trial})^n$.

Birthdays and matching

• All different: $\dfrac{^{365}P_n}{365^n}$. A shared birthday is the complement. For $n = 23$ it is about $0.507$.
• Derangements: $D_n = n!\displaystyle\sum_{k=0}^{n}\dfrac{(-1)^k}{k!}$ and $D_n = (n - 1)(D_{n-1} + D_{n-2})$.
• No match: $\dfrac{D_n}{n!}$, close to $\dfrac{1}{e}$. Exactly $r$ matches: $\dfrac{^nC_r \times D_{n-r}}{n!}$.

Geometric probability and simulation

• On a line: $\dfrac{\text{good length}}{\text{total length}}$. In a square: $\dfrac{\text{good area}}{\text{total area}}$.
• Two friends: both wait $w$ minutes in an hour: $1 - \left(\dfrac{60 - w}{60}\right)^2$.
• Simulation: it estimates. The usual error is about $\sqrt{\dfrac{p(1 - p)}{N}}$.

A worked example

Two dice are thrown. Find the probability that the total is at most $5$, and then the probability that it is at least $9$.

1. At most 5. The totals $2$, $3$, $4$, $5$ have $1$, $2$, $3$, $4$ ways, so $10$ outcomes and $\dfrac{10}{36} = \dfrac{5}{18}$.
2. At least 9. The totals $9$, $10$, $11$, $12$ have $4$, $3$, $2$, $1$ ways, so $10$ outcomes and $\dfrac{5}{18}$ again. This is the symmetry of the list.

Practice with new numbers

A program to try

The program finds the chance of a total of $7$ with two dice and of exactly three heads in six tosses. Then it estimates the chance that a rod of length $12$ cut at a random point has both pieces at least $3$ long.

Mistakes to avoid

• A total as one outcome. A total of $7$ is six outcomes.
• Forgetting the places of the successes. Exactly $k$ in $n$ trials needs $^nC_k$.
• A length for two random numbers. Use the square and an area.
• Calling an estimate exact. A simulation estimates.

Check yourself

1. How many of the $36$ outcomes of two dice give a total of $6$?

2. A fair coin is tossed $5$ times. Find the probability of exactly two heads.

3. Three letters are put into three envelopes at random. Find the probability that exactly one is correct.

4. A bus comes every $10$ minutes. Find the probability that you wait at most $4$ minutes.

5. A simulation of $2500$ trials gives an estimate near $0.5$. About how large is the usual error?

Sources

• OpenStax, Introductory Statistics, Chapter 3, section on terminology, and Chapter 5, section on the uniform distribution.
• OpenStax, Precalculus 2e, Chapter 11, sections on counting principles and on probability.