Question

6.6 Suppose a poll is taken to determine if voters support the President. The following data represent support and gender. What is the probability that a person selected at random: a. If a male, favors the President? b. Is a male and is undecided? c. Favors the President given that he/she is undecided? d. Favors the President or is a female?

Answer 1

a. The probability that a person selected at random, if a male, favors the President is 0.6.

b. The probability that a person selected at random is a male and is undecided is 0.15.

c. The probability that a person favors the President given that he/she is undecided is 0.4.

d. The probability that a person selected at random favors the President or is a female is 0.55.

Step-by-step explanation:

The probability of a male favoring the President is found by dividing the number of males who favor the President (3,000) by the total number of males (5,000), resulting in a probability of 0.6 (3,000/5,000 = 0.6). The probability of a male being undecided is given as 0.15 (5,000 * 0.15 = 750).

To determine the probability that a person favors the President given that they are undecided, we use the conditional probability formula: Probability (Favoring the President | Undecided) = Probability (Favoring the President and Undecided) / Probability (Undecided). With 750 individuals undecided and 3000 favoring the President, the probability is 0.4 (3000/750 = 0.4).

Lastly, to find the probability that a person selected at random favors the President or is a female, we add the probabilities of favoring the President and being a female and then subtract the intersection (favors the President and is a female) to avoid double counting. Thus, it's 0.55 ((3000 + 2000 - 1000) / 10,000 = 0.55).

Answer 2

The probabilities are:

a -
`(295)/(700)`, b -
`(150)/(700)`, c -
`(0)/(180)`, and d -
`(490)/(700)` or 0.7.

To calculate the probabilities, we can use the given data. Let's define the following events:

- Event A: Favoring the President

- Event U: Being Undecided

- Event M: Being Male

- Event F: Being Female

- P(A): Probability of favoring the President

- P(U): Probability of being undecided

- P(M): Probability of being male

- P(F): Probability of being female

Given data:

P(A) =
`\frac{\text{Number of favorable responses}}{\text{Total number of responses}}` =
`(295)/(700)`

P(U) =
`\frac{\text{Number of undecided responses}}{\text{Total number of responses}}` =
`(180)/(700)`

P(M) =
`\frac{\text{Number of male responses}}{\text{Total number of responses}}` =
`(400)/(700)`

P(F) =
`\frac{\text{Number of female responses}}{\text{Total number of responses}}` =
`(300)/(700)`

a. Probability that a person selected at random, if a male, favors the President:

P(A | M) =
`\frac{\text{Number of favorable responses from males}}{\text{Total number of male responses}}` =
`(190)/(400)`

b. Probability that a person selected at random is a male and is undecided:

P(M
`\cap` U) =
`\frac{\text{Number of males who are undecided}}{\text{Total number of responses}}` =
`(150)/(700)`

c. Probability that a person selected at random favors the President given that he/she is undecided:

P(A | U) =
`\frac{\text{Number of favorable responses from those who are undecided}}{\text{Total number of undecided responses}}` =
`(0)/(180)`

(The probability is undefined in this case since there are no favorable responses among the undecided individuals.)

d. Probability that a person selected at random favors the President or is a female:

P(A
`\cup` F) = P(A) + P(F) - P(A
`\cap` F)

P(A
`\cap` F) =
`\frac{\text{Number of favorable responses from females}}{\text{Total number of responses}}` =
`(105)/(700)`

P(A
`\cup` F) =
`(295)/(700)` +
`(300)/(700)` -
`(105)/(700)`

=
`(295 + 300 - 105)/(700)`

=
`(490)/(700)`

=
`(7)/(10)`

= 0.7

The given question is incomplete, but the complete question can be:

Suppose a poll is taken to determine if voters support the President. The following data represent support and gender. (refer image).

What is the probability that a person selected at random:

a. If a male, favors the President?

b. Is a male and is undecided?

c. Favors the President given that he/she is undecided?

d. Favors the President or is a female?