Questions: Kristina Karganova invites 14 relatives to a party: her mother, 4 aunts, 3 uncles, 2 brothers, 1 male cousin, and 3 female cousins. If the chances of any one guest arriving first are equally likely, find the probabilities that the first guest to arrive is as follows (a) A brother or an uncle (b) A brother or a cousin (c) A brother or her mother (a) Find the probability that the first guest to arrive is a brother or an uncle.

Kristina Karganova invites 14 relatives to a party: her mother, 4 aunts, 3 uncles, 2 brothers, 1 male cousin, and 3 female cousins. If the chances of any one guest arriving first are equally likely, find the probabilities that the first guest to arrive is as follows
(a) A brother or an uncle
(b) A brother or a cousin
(c) A brother or her mother
(a) Find the probability that the first guest to arrive is a brother or an uncle.

Transcript text: Kristina Karganova invites 14 relatives to a party: her mother, 4 aunts, 3 uncles, 2 brothers, 1 male cousin, and 3 female cousins. If the chances of any one guest arriving first are equally likely, find the probabilities that the first guest to arrive is as follows (a) A brother or an uncle (b) A brother or a cousin (c) A brother or her mother (a) Find the probability that the first guest to arrive is a brother or an uncle.

Solution

Solution Steps

To solve this problem, we need to calculate the probability of specific events occurring when all outcomes are equally likely. We will determine the total number of guests and then count the number of guests that fit each specified category. The probability of each event is the ratio of the number of favorable outcomes to the total number of outcomes.

(a) To find the probability that the first guest to arrive is a brother or an uncle, we count the number of brothers and uncles and divide by the total number of guests.

(b) To find the probability that the first guest to arrive is a brother or a cousin, we count the number of brothers and cousins and divide by the total number of guests.

(c) To find the probability that the first guest to arrive is a brother or her mother, we count the number of brothers and add one for the mother, then divide by the total number of guests.

Step 1: Total Guests

The total number of guests invited to the party is given as \( 14 \).

Step 2: Probability of a Brother or an Uncle

To find the probability that the first guest to arrive is a brother or an uncle, we calculate: \[ \text{Number of brothers} = 2, \quad \text{Number of uncles} = 3 \] Thus, the total number of favorable outcomes is: \[ 2 + 3 = 5 \] The probability is then: \[ P(\text{Brother or Uncle}) = \frac{5}{14} \approx 0.3571 \]

Step 3: Probability of a Brother or a Cousin

Next, we calculate the probability that the first guest to arrive is a brother or a cousin. The total number of cousins is: \[ \text{Number of male cousins} = 1, \quad \text{Number of female cousins} = 3 \] Thus, the total number of cousins is: \[ 1 + 3 = 4 \] The total number of favorable outcomes is: \[ 2 + 4 = 6 \] The probability is then: \[ P(\text{Brother or Cousin}) = \frac{6}{14} \approx 0.4286 \]

Step 4: Probability of a Brother or Her Mother

Finally, we calculate the probability that the first guest to arrive is a brother or her mother. The total number of favorable outcomes is: \[ 2 + 1 = 3 \] The probability is then: \[ P(\text{Brother or Mother}) = \frac{3}{14} \approx 0.2143 \]

Final Answer

The probabilities are as follows:

(a) \( P(\text{Brother or Uncle}) \approx 0.3571 \)
(b) \( P(\text{Brother or Cousin}) \approx 0.4286 \)
(c) \( P(\text{Brother or Mother}) \approx 0.2143 \)

Thus, the final answers are: \[ \boxed{P(\text{Brother or Uncle}) \approx 0.3571} \] \[ \boxed{P(\text{Brother or Cousin}) \approx 0.4286} \] \[ \boxed{P(\text{Brother or Mother}) \approx 0.2143} \]

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