An urn contains 2 red and 3 white balls. A ball is drawn, and then it and another ball of the same color are placed back in the urn. Finally, a second ball is drawn. Given that the second ball drawn is white, what is the conditional probability that the first ball drawn was red?
$\left( {{1}\over{3}}\right) \left( {{3}\over{5}}\right) \left/\left[ \lef... ...over{3}}\right) \left( {{3}\over{5}}\right) \right]\right.= {{1}\over{2}}$ $\left( {{1}\over{2}}\right) \left( {{2}\over{5}}\right) \left/\left[ \lef... ...over{3}}\right) \left( {{3}\over{5}}\right) \right]\right.= {{1}\over{2}}$ $\left( {{2}\over{3}}\right) \left( {{3}\over{5}}\right) \left/\left[ \lef... ...over{3}}\right) \left( {{3}\over{5}}\right) \right]\right.= {{2}\over{3}}$ $\left( {{1}\over{2}}\right) \left( {{2}\over{5}}\right) \left/\left[ \lef... ...over{3}}\right) \left( {{3}\over{5}}\right) \right]\right.= {{1}\over{3}}$
Suppose that $P(A) = {{2}\over{5}}$ , $P(B) = {{1}\over{5}}$ , and that and are independent. Then find $P(A\cup B)$ .
$\displaystyle {{2}\over{25}}$ $\displaystyle {{3}\over{5}}$ $\displaystyle {{13}\over{25}}$ $\displaystyle {{17}\over{25}}$
Urn 1 has 5 red balls and 4 white balls, and urn 2 has 3 red balls and 2 white balls. A fair coin is tossed; if it lands heads up, a ball is drawn from urn 1, and otherwise, a ball is drawn from urn 2. Given that a red ball is drawn, what is the conditional probability that the coin landed heads up?
$\displaystyle {{1}\over{2}}$ $\displaystyle {{27}\over{52}}$ $\displaystyle {{25}\over{52}}$ $\displaystyle {{5}\over{9}}$
An accident-prone person will have an accident with probability $\displaystyle {{1}\over{2}}$ , whereas this probability is only $\displaystyle {{1}\over{5}}$ for a non-accident-prone person. Suppose that a new policyholder is accident prone with probability $\displaystyle {{1}\over{5}}$ . Given that the new policyholder had an accident, what is the conditinal probability that the policyholder is an accident-prone person?
$\displaystyle {{1}\over{11}}$ $\displaystyle {{5}\over{13}}$ $\displaystyle {{10}\over{11}}$ $\displaystyle {{8}\over{13}}$
Urn 1 has 2 red balls and 3 white balls, and urn 2 has 5 red balls and 3 white balls. A fair coin is tossed; if it lands heads up, a ball is drawn from urn 1, and otherwise, a ball is drawn from urn 2. What is the probability that a red ball is drawn?
$\displaystyle {{5}\over{8}}$ $\displaystyle {{2}\over{5}}$ $\displaystyle {{39}\over{80}}$ $\displaystyle {{41}\over{80}}$
An urn contains 2 red and 3 white balls. A ball is drawn, and then it and another ball of the same color are placed back in the urn. Finally, a second ball is drawn. What is the probability that the second ball drawn is white?
$\left( {{1}\over{3}}\right) \left( {{2}\over{5}}\right) + \left( {{1}\over{2}}\right) \left( {{3}\over{5}}\right) = {{2}\over{5}}$ $\left( {{1}\over{2}}\right) \left( {{2}\over{5}}\right) + \left( {{1}\over{3}}\right) \left( {{3}\over{5}}\right) = {{2}\over{5}}$ $\left( {{2}\over{3}}\right) \left( {{2}\over{5}}\right) + \left( {{1}\over{2}}\right) \left( {{3}\over{5}}\right) = {{17}\over{30}}$ $\left( {{1}\over{2}}\right) \left( {{2}\over{5}}\right) + \left( {{2}\over{3}}\right) \left( {{3}\over{5}}\right) = {{3}\over{5}}$
An accident-prone person will have an accident with probability $\displaystyle {{3}\over{5}}$ , whereas this probability is only $\displaystyle {{3}\over{10}}$ for a non-accident-prone person. Suppose that a new policyholder is accident prone with probability $\displaystyle {{1}\over{5}}$ . Then find the probability that a new policyholder will have an accident.
$\displaystyle {{6}\over{25}}$ $\displaystyle {{9}\over{25}}$ $\displaystyle {{3}\over{25}}$ $\displaystyle {{27}\over{50}}$
Suppose that $P(A) = {{7}\over{10}}$ , $P(B) = {{3}\over{5}}$ , and that and are independent. Then find $P(A^c\cup B^c)$ .
$\displaystyle {{29}\over{50}}$ $\displaystyle {{21}\over{50}}$ $\displaystyle -{{3}\over{10}}$ $\displaystyle {{22}\over{25}}$
A couple has two children. What is the conditional probability that both are girls given that at least one of them is a girl?
$\displaystyle {{1}\over{3}}$ $\displaystyle {{1}\over{2}}$ $\displaystyle {{1}\over{4}}$ 1
A die is rolled 6 times. If the face numbered is the outcome on the -th roll, we say “a match has occurred.” You win if at least one match occurs during the 6 trials. Find the probability that you wins.
$1 - \left(\frac{1}{6}\right)^{ 6}$ $1 - \left(\frac{5}{6}\right)^{ 6}$ $\left(\frac{5}{6}\right)^{ 6}$ $\left(\frac{1}{6}\right)^{ 6}$

Department of Mathematics
Last modified: 2024-12-24