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Conditional and binomial probability
Conditional probability, the multiplication rule, independent events and binomial probability. 9 problems with step-by-step solutions.
Solve each problem in your notebook first, then open the solution.
Problem 1 Conditional probability
\(P(A\mid B) = \dfrac{P(A\cap B)}{P(B)}\).
a) A die is rolled twice. Find the probability that the sum is 8, if one of the numbers is 3.
b) Two dice are rolled. Find the probability that the sum is greater than 8, if the first die shows 5.
c) A die shows a number greater than 2. Find the probability that it is even.
Show solution
a) Outcomes with at least one 3: 11. Among them sum 8: (3,5), (5,3): \(\frac{2}{11}\).
b) Second die 4, 5 or 6: \(\frac36 = \frac12\).
c) \(\{3, 4, 5, 6\}\), even: 2. \(\frac12\).
Problem 2 Two-way table
In a group of 60 students, 25 study chemistry, 30 study biology and 10 study both. One student is chosen.
a) Find the probability that the student studies only biology.
b) Find the probability that the student studies chemistry, if we know that he studies biology.
c) Find the probability that he does not study biology, if he studies chemistry.
Show solution
a) \(\frac{30 - 10}{60} = \frac13\).
b) \(\frac{10}{30} = \frac13\).
c) \(\frac{25 - 10}{25} = \frac35\).
Problem 3 Exams
In a class of 30 students, 18 passed math, 15 passed physics and 4 failed both. A student who passed physics is chosen. Find the probability that he also passed math.
Show solution
Passed at least one: 26. Both: \(18 + 15 - 26 = 7\). \(P = \frac{7}{15}\).
Problem 4 Multiplication rule
\(P(A\cap B) = P(B)\cdot P(A\mid B)\).
a) Two cards are drawn without replacement. Find the probability that the first is a jack and the second is a queen.
b) A box has 5 white, 4 red and 3 blue balls. A ball is drawn and replaced, three times. Find the probability of white, then red, then blue.
c) Three balls are drawn without replacement from a box with 4 red, 5 yellow and 6 green balls. Find the probability that all three are green.
Show solution
a) \(\frac{4}{52}\cdot\frac{4}{51} = \frac{16}{2652} = \frac{4}{663}\).
b) \(\frac{5}{12}\cdot\frac{4}{12}\cdot\frac{3}{12} = \frac{60}{1728} = \frac{5}{144}\).
c) \(\frac{6}{15}\cdot\frac{5}{14}\cdot\frac{4}{13} = \frac{120}{2730} = \frac{4}{91}\).
Problem 5 Independent events
a) A die is rolled and a coin is tossed. Find the probability of a 6 and a tail.
b) Two dice are rolled twice. Find the probability that the sum is 7 both times.
c) 4% of brand A phones and 3% of brand B phones are faulty. You buy one of each. Find the probability that both are faulty, and that at least one is faulty.
d) A coin is tossed and a die is rolled. Find the probability of a head or an odd number.
Show solution
a) \(\frac16\cdot\frac12 = \frac{1}{12}\).
b) \(\left(\frac16\right)^2 = \frac{1}{36}\).
c) Both: \(0.04\cdot 0.03 = 0.0012\). At least one: \(1 - 0.96\cdot 0.97 = 0.0688\).
d) \(\frac12 + \frac12 - \frac14 = \frac34\).
Problem 6 Two urns
The first urn has 4 white and 2 black balls; the second has 3 white and 5 black. A ball is taken from the first urn and put into the second without looking. Then a ball is drawn from the second urn. Find the probability that it is white.
Show solution
If a white ball moved (probability \(\frac46\)): the second urn has 4 white of 9. If a black ball moved (\(\frac26\)): 3 white of 9.
\(P = \frac46\cdot\frac49 + \frac26\cdot\frac39 = \frac{16 + 6}{54} = \frac{11}{27}\).
Problem 7 Mixed problems
a) A box has 3 green and 7 white marbles. Two are drawn one after another without replacement. Find the probability of two green marbles.
b) A box has 6 disks of brand A and 3 of brand B. Two are taken. Find the probability that they are the same brand.
c) There are 5 married couples in a room. A man and a woman are chosen at random. Find the probability that they are married to each other.
d) A keyboard has 50 keys and a cat presses 3 keys at random. Find the probability that it types the word CAT.
Show solution
a) \(\frac{3}{10}\cdot\frac29 = \frac{1}{15}\).
b) \(\frac{C(6, 2) + C(3, 2)}{C(9, 2)} = \frac{15 + 3}{36} = \frac12\).
c) \(\frac15\).
d) \(\left(\frac{1}{50}\right)^3 = \frac{1}{125000}\).
Problem 8 Binomial probability
\(P(x \text{ successes in } n \text{ trials}) = \dbinom{n}{x}s^x f^{\,n - x}\).
a) A fair coin is tossed 6 times. Find the probability of exactly 4 heads.
b) A die is rolled 4 times. Find the probability that a 6 comes up exactly twice.
c) A die is rolled 5 times. Find the probability that a number less than 3 comes up exactly three times.
Show solution
a) \(\binom64\left(\frac12\right)^6 = \frac{15}{64}\).
b) \(\binom42\left(\frac16\right)^2\left(\frac56\right)^2 = \frac{6\cdot 25}{1296} = \frac{25}{216}\).
c) \(s = \frac13\): \(\binom53\left(\frac13\right)^3\left(\frac23\right)^2 = \frac{10\cdot 4}{243} = \frac{40}{243}\).
Problem 9 Binomial probability
a) A player scores a penalty with probability \(\frac34\). Find the probability of at least 4 goals in 5 penalties.
b) A basketball player scores 80% of free throws. His team is 1 point behind and he has 2 free throws. Find the probability that his team wins (each throw is 1 point; a tie is not a win).
c) Four cards are drawn with replacement. Find the probability of exactly two hearts.
Show solution
a) \(\binom54\left(\frac34\right)^4\frac14 + \left(\frac34\right)^5 = \frac{405}{1024} + \frac{243}{1024} = \frac{648}{1024} = \frac{81}{128}\).
b) He must score both: \(0.8^2 = 0.64\).
c) \(\binom42\left(\frac14\right)^2\left(\frac34\right)^2 = \frac{6\cdot 9}{256} = \frac{27}{128}\).