Do probability answers need to be reduced?

Yes. Give probabilities as fractions in lowest terms; unreduced fractions are marked incorrect on this site.

Math A

Probability (Math A): Practice Problems and Methods

Count so that every outcome is equally likely. Treating balls of the same color as different is the standard trick.

Basic, Standard: Grade 10 · Term 2 / Advanced: Grade 12 · Exam prep

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Basics and complements

The probability of an event $A$ is $P(A) = \dfrac{n(A)}{n(U)}$ . Two dice give 36 outcomes; drawing two balls from a bag of $n$ gives ${}_{n}\mathrm{C}_{2}$ .

Unions and complements

P(A \cup B) = P(A) + P(B) - P(A \cap B),\qquad P(\overline{A}) = 1 - P(A)

"At least one" is easiest with the complement. Example: the probability of at least one 6 with three dice is $1 - \left(\frac{5}{6}\right)^3 = \frac{91}{216}$ .

Independent and repeated trials

Probabilities of independent trials multiply. If an event has probability $p$ in one trial, the probability it happens exactly $r$ times in $n$ trials is

{}_{n}\mathrm{C}_{r}\,p^r(1 - p)^{n - r}

In "first to win 3 games" problems, remember that the deciding game is the last one: count the earlier games as repeated trials.

Conditional probability and expected value

Conditional probability and the multiplication rule

P_A(B) = \frac{P(A \cap B)}{P(A)},\qquad P(A \cap B) = P(A)P_A(B)

For "given that the ball is red, what is the probability it came from box A?", find $P(\text{red})$ by splitting over the boxes and apply the formula above.

The expected value is the sum of (value) × (probability of that value); it is the long-run average amount.

Q1BasicGr 10 · Term 2Probability (Math A)

Two cards are drawn at the same time from 10 cards numbered 1 to 10. Find the probability that the product is odd.

Answer

\frac{2}{9}

Solution

The product is odd when both are odd: ${}_{5}\mathrm{C}_{2} = 10$ ways.
Hence
$\frac{10}{45} = \frac{2}{9}$

Q2StandardGr 10 · Term 2Probability (Math A)

A die is thrown 5 times. Find the probability that a 1 or a 2 comes up exactly 3 times.

Answer

\frac{40}{243}

Solution

By the formula for repeated trials,
${}_{5}\mathrm{C}_{3} \left(\frac{1}{3}\right)^{3} \left(\frac{2}{3}\right)^{2} = \frac{40}{243}$

Q3AdvancedExam prepProbability (Math A)

2 dice are thrown. Find the probability that the smallest number is 5.

Answer

\frac{1}{12}

Solution

Take "min at least 5" minus "min at least 6":
$\frac{2^{2} - 1^{2}}{6^{2}} = \frac{1}{12}$

Q1BasicGr 10 · Term 2Probability (Math A)

Two dice (one large, one small) are thrown. Find the probability that the product is a multiple of 4.

Q2StandardGr 10 · Term 2Probability (Math A)

A bag has 5 red and 4 white balls. Two balls are drawn one at a time without replacement. Given that the first is red, find the conditional probability that the second is also red.

Q3AdvancedExam prepProbability (Math A)

A bag has 4 red and 5 white balls. Two balls are drawn one at a time without replacement. Given that the second ball is red, find the conditional probability that the first ball was also red.

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Counting: permutations and combinations

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