91Ó°ÊÓ

A given pair has an equal chance of having a boy or a girl.

The number X symbolizes the number of girls in a household of four.

No girl, one girl, two girls, three girls, or four girls are all options for a family with four children.

As a result, the random variable X's possible values are 0, 1, 2, 3, and 4.

Calculation:

A family with four children's sample space is,

S = {BBBB, GBBB, BGBB, BBGB, BBBG, GGBB, BGGB, BBGG, GBBG, BGBG, GBGB, BGGG, GBGG, GGBG, GGGB, GGGG }

The potential values of random variable X are 0, 1, 2, 3 and 4.

P(X = 0) = P{BBBB} = 1/16

P(X =1) = P{GBBB, BGBB, BBGB, BBBG} = 4/16 = 1/4

P(X = 2) = P(GGBB, BGGB, BBGG, GBBG, BGBG, GBGB) = 6/16 = 3/8

P(X = 3) = P{BGGG, GBGG, GGBG, GGGB} = 4/16 = 1/4

P(X = 4) = P{GGGG} = 1/16

Hence the probability distribution of random variable X is,

The occurrence "exactly two girls" implies that the random variable value must be 2.

As a result, the random variable notation for exactly two girls is as follows:

X = 2,

Its probability is,

P(X = 2)=P(GGBB, BGGB, BBGG, GBBG, BGBG, GBGB)

= 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16

= 6/16 = 3/8

The condition "at least two girls" requires that the random variable value be more than or equal to 2.

As a result, at least two females' random variable notation is,

X ≥ 2

Its probability is,

P(X ≥ 2) = P(X = 2) + P(X = 3) + P(X = 4)

= 6/16 + 4/16 + 1/16

= 11/16

The event "at most two girls" requires that the random variable value be less than or equal to 2.

As a result, the random variable notation for a maximum of two girls is,

X ≤ 2

Its probability is,

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

= 1/16 + 4/16 + 6/16

= 11/16

The event "between one and three girls, inclusive" indicates that the random variable value must be between one and three, including one and three.

As a result, the random variable notation for the event involving one to three ladies is,

1 ≤ X ≤ 3

Its probability is,

P(1 ≤ X ≤ 3) = P(X = 1) + P(X = 2) + P(X = 3)

= 4/16 + 6/16 + 4/16

= 14/16 = 7/8

The random variable value must be 0 (4 boys) or 4 (4 girls) if the event "children all of the same gender" occurs (4 girls).

As a result, the event children's random variable notation for everyone of the same gender is,

X = 0 or X = 4

Its probability is,

P(X=0 or X=4) = P(X = 0) + P(X = 4)

= 1/16 + 1/16

= 2/16 = 1/8