Chapter 2 Probability - 2.4 Conditional Probability - Questions - Page 37: 11

a)The probability that someone likes neither is 5/100. b)The probability that someone likes exactly one is 70/100. c)The probability that someone likes at least one is 95/100. d)The probability that someone likes at most one is 75/100. e)The probability that someone likes exactly one given that he or she likes at least one is 70/95. f)The proportion of that like both is 25/95. g)The proportion of that like “B” is 25/95.

(a) Let A be the event that voters like candidate “A” and let B be the event that voters like candidate “B”. \[P(A)=\frac{65}{100}\] \[P(B)=\frac{55}{100}\] \[P(A\cap B)=P(like\text{ both})=\frac{25}{100}\] We have to find the probability of that someone likes neither that means \[P{{(A\cup B)}^{C}}\] \[\begin{align} & P{{(A\cup B)}^{C}}=1-P(A\cup B) \\ & P{{(A\cup B)}^{C}}=1-[P(A)+P(B)-P(A\cap B)] \\ & P{{(A\cup B)}^{C}}=1-\left[ \frac{65}{100}+\frac{55}{100}-\frac{25}{100} \right] \\ & P{{(A\cup B)}^{C}}=1-\frac{95}{100} \\ & P{{(A\cup B)}^{C}}=\frac{5}{100} \\ \end{align}\] Therefore, the probability that someone likes neither is 5/100. (b) \[P(A)=\frac{65}{100}\] \[P(B)=\frac{55}{100}\] \[P(A\cap B)=P(like\text{ both})=\frac{25}{100}\] The probability of a voter liking only candidate “A” is \[=\frac{65}{100}-\frac{25}{100}=\frac{40}{100}\] The probability of a voter liking only candidate “B” is \[=\frac{55}{100}-\frac{25}{100}=\frac{30}{100}\] We have to find the probability that someone likes exactly one: \[\begin{align} & =P(\text{voter like exactly candidate ''A'')}+P(\text{voter like exactly candidate ''B'')} \\ & \text{=}\frac{40}{100}+\frac{30}{100} \\ & =\frac{70}{100} \\ \end{align}\] Therefore, the probability of that someone likes exactly one is 70/100. (c) \[P(A)=\frac{65}{100}\] \[P(B)=\frac{55}{100}\] \[P(A\cap B)=P(like\text{ both})=\frac{25}{100}\] We have to find the probability of that someone likes at least one that means \[P(A\cup B)\] \[\begin{align} & P(A\cup B)=P(A)+P(B)-P(A\cap B) \\ & P(A\cup B)=\frac{65}{100}+\frac{55}{100}-\frac{25}{100} \\ & P(A\cup B)=\frac{95}{100} \\ \end{align}\] Therefore the probability of that someone likes at least one is 95/100. (d) Given, \[P(A\cap B)=P(like\text{ both})=\frac{25}{100}\] \[\begin{align} & P{{(A\cap B)}^{C}}=1-P(A\cap B) \\ & P{{(A\cap B)}^{C}}=1-\frac{25}{100} \\ & P{{(A\cap B)}^{C}}=\frac{75}{100} \\ \end{align}\] (e) Let C be the event that someone likes at most one and D be the even that he or she likes at least one. We know that, \[P\left( \text{someone likes exactly one} \right)=P(C)=\frac{70}{100}\] \[P\left( \text{someone likes at least one} \right)=P(D)=\frac{95}{100}\] We have to find the probability of that someone likes exactly one given that he or she likes at least one; that means $P(C|D)$ \[\begin{align} & P(C|D)=\frac{P(C\cap D)}{P(D)} \\ & P(C|D)=\frac{P(C)}{P(D)} \\ & P(C|D)=\frac{{}^{70}/{}_{100}}{{}^{95}/{}_{100}} \\ & P(C|D)=\frac{70}{95} \\ \end{align}\] (f) We know that, \[P(like\text{ both})=\frac{25}{100}\] \[P\left( \text{someone likes at least one} \right)=\frac{95}{100}\] So: \[\begin{align} & P(like\text{ }both|atleast\text{ }one)=\frac{P(like\text{ }both\text{ }and\text{ }atleast\text{ }one)}{P(atleast\text{ }one)} \\ & P(like\text{ }both|atleast\text{ }one)=\frac{P(like\text{ }both)}{P(atleast\text{ }one)} \\ & P(like\text{ }both|atleast\text{ }one)=\frac{{}^{25}/{}_{100}}{{}^{95}/{}_{100}} \\ & P(like\text{ }both|atleast\text{ }one)=\frac{25}{95} \\ \end{align}\] (g) We know that, The probability of a voter liking only candidate “B” is \[=\frac{55}{100}-\frac{25}{100}=\frac{30}{100}\] \[\begin{align} & P\left( \text{do not like }A \right)=1-P(A) \\ & P\left( \text{do not like }A \right)=1-\frac{65}{100} \\ & P\left( \text{do not like }A \right)=\frac{35}{100} \\ \end{align}\] We have to find the probability that one likes “B” given that he or she does not like “A”: \[\begin{align} & P(like\text{ }B|do\text{ }not\text{like }A)=\frac{P(llike\text{ }B\text{ }and\text{ }do\text{ }not\text{like }A)}{P(do\text{ }not\text{like }A)} \\ & P(like\text{ }B|do\text{ }not\text{like }A)=\frac{P(like\text{ }B)}{P(do\text{ }not\text{like }A)} \\ & P(like\text{ }B|do\text{ }not\text{like }A)=\frac{{}^{30}/{}_{100}}{{}^{35}/{}_{100}} \\ & P(like\text{ }B|do\text{ }not\text{like }A)=\frac{30}{35} \\ \end{align}\]