A certain couple is equally likely to have either a boy or a...

A certain couple is equally likely to have either a boy or a girl. If the family has four children, let $X$ denote the number of girls. a. Identify the possible values of the random variable $X .$ b. Determine the probability distribution of $X$. (Hint: There are 16 possible equally likely outcomes. One is GBBB, meaning the first born is a girl and the next three born are boys.) Use random-variable notation to represent each of the following events. Also use the special addition rule and the probability distribution obtained in part (b) to determine each event's probability. The couple has c. exactly two girls. d. at least two girls. e. at most two girls. f. between one and three girls, inclusive. g. children all of the same gender.

A certain couple is equally likely to have either a boy or a girl. If the family has four children, let $X$ denote the number of girls.
a. Identify the possible values of the random variable $X .$
b. Determine the probability distribution of $X$. (Hint: There are 16 possible equally likely outcomes. One is GBBB, meaning the first born is a girl and the next three born are boys.) Use random-variable notation to represent each of the following events. Also use the special addition rule and the probability distribution obtained in part (b) to determine each event's probability. The couple has
c. exactly two girls.
d. at least two girls.
e. at most two girls.
f. between one and three girls, inclusive.
g. children all of the same gender.

Key Concepts

Equally Likely Outcomes

When outcomes of an experiment are equally likely, each outcome has the same probability of occurrence. This concept simplifies the calculation of event probabilities by allowing one to compute probability as the ratio of the number of favorable outcomes to the total number of outcomes.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. Its probability mass function, given by P(X = k) = C(n, k) * p^k * (1-p)^(n-k), provides a powerful tool for dealing with binary outcomes in probability problems.

Addition Rule

The addition rule in probability is used to compute the probability of the occurrence of at least one of several mutually exclusive events by summing their individual probabilities. This rule is particularly useful when determining the probability of compound events that do not overlap.

Random Variable

A random variable is a numerical function defined on a sample space that assigns a number to each outcome, which allows the analysis of probabilistic events in numerical terms. It is essential for translating real-world situations into mathematical models for probability calculations.

Discrete Probability Distribution

A discrete probability distribution specifies the probability that a discrete random variable takes on each of its possible values. This type of distribution is characterized by a set of distinct outcomes, where the sum of all the individual probabilities equals 1.

Transcript

00:01 So we know that the family has four children in this case.

00:04 If x represents the number of girls, then the possible number of girls would obviously be 0, 1, 2, 3, and 4.

00:13 Then if we let g represent a girl and b represent a boy, there's all the possible combinations of the four children, which is all boys, all boys, and then lastly a girl.

00:27 And we can go through all these outcomes.

00:28 And there are 16 outcomes.

00:32 In those outcomes, there's one outcome with no girl.

00:40 Then there's four with one girl.

00:46 There are six with two girls.

00:51 There are four with three girls.

00:55 And once again, we have one with four girls.

01:01 So the probability is the number of favorable outcomes divided by the number of possible outcomes.

01:06 So we know that p x0 is going to be one because there is one possible chance of that over the 16.

01:16 That's going to give a 0 .0625.

01:18 We go through this, we get x0.

01:24 It's going to be 0 .25.

01:27 That's a possibility of one girl, possibility, the probability of, sorry, of two girls is 0 .375.

01:37 Probability of three girls is going to be 0 .25.

01:42 And then we see again the probability of four girls is going to be the same as the first one, so 0 .0625.

01:51 And then for part c, we want to know the event that there are at least two girls...

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