Conditional probability from exact representations lesson | Year 10 Mathematics

A membership	\(B\)	\(B^{\prime}\)	Total
\(A\)	?	?	16
\(A^{\prime}\)	6	6	12
Total	12	16	28

A membership	\(B\)	\(B^{\prime}\)	Total
\(A\)	\(\frac{3}{10}\)	\(\frac{3}{10}\)	\(\frac{3}{5}\)
\(A^{\prime}\)	\(\frac{13}{40}\)	\(\frac{3}{40}\)	\(\frac{2}{5}\)
Total	\(\frac{5}{8}\)	\(\frac{3}{8}\)	\(1\)

A membership	\(B\)	\(B^{\prime}\)	Total
\(A\)	7	10	17
\(A^{\prime}\)	11	17	28
Total	18	27	45

A membership	\(B\)	\(B^{\prime}\)	Total
\(A\)	\(\frac{6}{35}\)	\(\frac{8}{35}\)	\(\frac{2}{5}\)
\(A^{\prime}\)	\(\frac{16}{35}\)	\(\frac{1}{7}\)	\(\frac{3}{5}\)
Total	\(\frac{22}{35}\)	\(\frac{13}{35}\)	\(1\)

Conditional probability from known information

Use known information to restrict a sample space, distinguish both conditional directions, and recover joint probabilities

Identify the known group Words such as given, if, among and of can specify the group already known. Read the sentence carefully before choosing a denominator
Keep the overlap Within the known group, count only outcomes that also satisfy the target event \(\displaystyle \Pr(A\mid B)=\frac{\Pr(A\cap B)}{\Pr(B)}\)
Check the condition The conditioning event must have positive probability. A zero numerator can give zero; a zero denominator is undefined
Use matching units Divide a count by a count or a probability by a probability. Do not divide a count by a probability
Directions differ A given B and B given A generally use different denominators
Recover a joint probability Multiply the conditional probability by the probability of its condition \(\displaystyle \Pr(A\cap B)=\Pr(A\mid B)\Pr(B)\)

Worked example

Twenty equally likely cards include six blue-round, four blue-not-round, two not-blue-round and eight neither. Let A mean blue and B mean round. Find the probability of A given B, the probability of B given A, and the joint probability of A and B.

\(\displaystyle N(B)=6+2=8\)B is the condition for A given B.
\(\displaystyle \Pr(A\mid B)=\frac68=\frac34\)Only the six shared cards are favourable
\(\displaystyle \Pr(B\mid A)=\frac6{10}=\frac35\)Reversing the condition uses the ten blue cards
\(\displaystyle \Pr(A\cap B)=\frac6{20}=\frac3{10}\)The joint probability uses the whole collection

Common mistakes

Using the whole collection

✕\(\displaystyle \Pr(A\mid B)=\frac6{20}\)✓\(\displaystyle \Pr(A\mid B)=\frac68=\frac34\)

Restrict the denominator to B.

Reversing the condition

✕\(\displaystyle \Pr(A\mid B)=\frac6{10}\)✓\(\displaystyle \Pr(A\mid B)=\frac68\)

Read the event after the bar

Calling an impossible condition zero

✕\(\displaystyle \frac00=0\)✓\(\displaystyle \frac{0}{0}\text{ is undefined\)

The elementary conditional formula is undefined when its condition has probability zero

Difficulty levels

introRestrict a uniformly selected listed sample space

standardComplete count tables or use joint-probability tables

challengeCompare both directions or recover a joint probability

examSelect the formula orientation and calculate from exact data

Questions about this topic

What does given mean?+

It identifies information already known, so only outcomes consistent with it remain possible

Does of always mean multiply?+

No. Of the round cards describes a restricted group here. Interpret the sentence before choosing an operation

Can I use counts instead of probabilities?+

Yes, for uniform selection among the counted items. Divide the favourable count within the condition by the total count in that condition

Why do reverse directions differ?+

The shared numerator is the same, but the conditioning groups can have different sizes

Can a conditional probability be zero or one?+

Yes. It is zero when no outcome in the possible condition is favourable, and one when every such outcome is favourable

Can a condition with probability zero be used?+

Not in this elementary formula, because it would require division by zero

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 10 Mathematics, chapter 4: Probability

In this chapter

Describe and order probability values Single-step theoretical probability and complements 4A Probability scale and experiments: Experimental probability from observed frequencies Finite events, probability assignments and complements 4A Probability scale and experiments: Design and interpret a probability simulation 4B Venn diagrams and sets: Venn regions, set language and probability Compound events, the addition rule and mutual exclusivity Complete and use a two-way probability table Construct and complete two-way frequency tables Convert between Venn diagrams and two-way tables Conditional probability from exact representations 4D Conditional probability: Conditional probability on Venn diagrams Two-step sample-space tables with and without replacement Represent a multi-stage finite sample space Tree-diagram probability for two-stage experiments Multi-stage probability, multiplication and total probability 4G Independence and simulations: Independence on Venn diagrams Test exact independence 4G Independence and simulations: Design and interpret a probability simulation 4G Independence and simulations: Experimental probability from observed frequencies

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Conditional probability from exact representations

Contents

Restrict, then compare

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