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4.4 Conditional Probability
When additional information is known that will affect the probability of an outcome, we calculate what
is called a conditional probability.
𝑃(𝐴 ∩ 𝐡)
𝑃(𝐴|𝐡) =
𝑃(𝐡)
𝑃(𝐴|𝐡) is read β€œthe probability of event A, given that the event B has occurred”.
Once B is known, then the event set for A is restricted to those
outcomes in the set (𝐴 ∩ 𝐡). So the outcome set is reduced
from S to B.
Example 1
What is the probability of rolling a sum greater than 7 with two dice if it is known that
the first die rolled is a 3?
𝑃(π‘ π‘’π‘š > 7|1𝑠𝑑 π‘Ÿπ‘œπ‘™π‘™ 𝑖𝑠 3) =
2⁄
36
1
6
2 6
=
×
36 1
12
=
36
1
=
3
𝑃(π‘ π‘’π‘š > 7 ∩ 1𝑠𝑑 π‘Ÿπ‘œπ‘™π‘™ 𝑖𝑠 3)
𝑃(1𝑠𝑑 π‘Ÿπ‘œπ‘™π‘™ 𝑖𝑠 3)
=
Example 2
If a family is chosen at random from the set of all families with exactly two children,
determine the probability that
a) The family has two boys given that at least one child is a boy
𝑃(2 π‘π‘œπ‘¦π‘ |π‘œπ‘›π‘’ 𝑖𝑠 π‘Ž π‘π‘œπ‘¦)
1st
2nd
𝑃(2 π‘π‘œπ‘¦π‘  ∩ π‘œπ‘›π‘’ 𝑖𝑠 π‘Ž π‘π‘œπ‘¦)
boy
BB
=
boy
𝑃(π‘œπ‘›π‘’ 𝑖𝑠 π‘Ž π‘π‘œπ‘¦)
girl
BG
4 outcomes
1⁄
4
𝑛(𝑆) = 4
=
3⁄
boy
GB
4
girl
1 4 1
= × =
GG
girl
4 3 3
b) The family has two boys if it is known that the first child is a boy
𝑃(2 π‘π‘œπ‘¦π‘ |1𝑠𝑑 𝑖𝑠 π‘Ž π‘π‘œπ‘¦)
𝑃(2π‘π‘œπ‘¦π‘  ∩ 1𝑠𝑑 𝑖𝑠 π‘Ž π‘π‘œπ‘¦) 1⁄4
=
=
2⁄
𝑃(1𝑠𝑑 𝑖𝑠 π‘Ž π‘π‘œπ‘¦)
4
1
2
=4×1=½
Since 𝑃(𝐡) × π‘ƒ(𝐴|𝐡) =
𝑃(𝐴∩𝐡)
𝑃(𝐡)
× π‘ƒ(𝐡), then 𝑃(𝐡) × π‘ƒ(𝐴|𝐡) = 𝑃(𝐴 ∩ 𝐡)
Multiplication Law for Conditional Probability
𝑃(𝐴 ∩ 𝐡) = 𝑃(𝐴|𝐡) × π‘ƒ(𝐡)
Example 3
If the first card drawn from a standard deck of cards is not replaced, what is the
probability of…
a) Drawing an ace on the first draw AND a jack on the second draw?
𝑃(2𝑛𝑑 π‘—π‘Žπ‘π‘˜ ∩ 1𝑠𝑑 π‘Žπ‘π‘’)
= 𝑃(2𝑛𝑑 π‘—π‘Žπ‘π‘˜|1𝑠𝑑 π‘Žπ‘π‘’) × π‘ƒ(1𝑠𝑑 π‘Žπ‘π‘’)
4
4
=
×
51 52
16
4
=
=
2652 663
b) Drawing a pair of eights?
𝑃(2𝑛𝑑 8 ∩ 1𝑠𝑑 8)
= 𝑃(2𝑛𝑑 8|1𝑠𝑑 8) × π‘ƒ(1𝑠𝑑 8)
3
4
12
1
=
×
=
=
51 52 2652 221
Homework p. 235 # 1, 2, 4, 6, 7, 9, 10
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