Question; Use a tree diagram to illustrate the possible outcomes for a couple having 4 children...

Question

Please Help, thank you so much.

1.    Use a tree diagram to illustrate the possible outcomes for a couple having 4 children, assuming that the probability of having a boy equals the probability of having a girl. What is the probability that the family will have all 4 boys or all 4 girls? Please show all work and explain.

Answer 1

The Probability Tree for all possible outcomes in this problem:

For one child, there are only two possible outcomes = either a boy or a girl. So there is a 50% probability for each outcome. But if there are 4 children, the Probability Tree will show you all possible outcomes in this problem:

[1st Child] [2nd child] [3rd child] [4th child]

| | | |

↓ ↓ ↓ ↓

⌈¯¯¯¯^0.5¯¯B

⌈¯¯¯¯^0.5¯¯¯¯B––_0.5– G

↑ ⌈¯¯¯¯^0.5¯¯B

⌈¯¯¯¯^0.5¯¯¯¯B––_0.5–––G––_0.5–G

|

| ⌈¯¯¯¯^0.5¯¯B

| ⌈¯¯¯¯^0.5¯¯¯¯B––_0.5––G

↑ ↑ ⌈¯¯¯¯^0.5¯¯B

⌈¯¯¯¯¯^0.5¯¯¯B¯¯¯^0.5¯¯¯¯¯G––_0.5–––G––_0.5–G

| ⌈¯¯¯¯^0.5¯¯B

| ⌈¯¯¯¯^0.5¯¯¯¯B––_0.5– G

| ↑ ⌈¯¯¯¯^0.5¯¯B

| ⌈¯¯¯¯^0.5¯¯¯¯B––_0.5–––G––_0.5–G

| |

| | ⌈¯¯¯¯^0.5¯¯B

| | ⌈¯¯¯¯^0.5¯¯¯¯B––_0.5––G

↑ ↑ ↑ ⌈¯¯¯¯^0.5¯¯B

^Start¯¯^0.5¯¯G¯¯¯^0.5¯¯¯¯¯G––_0.5–––G––_0.5–G

There are 2⁴=16 possible outcomes.

However, we only need all 4 boys

Start¯¯¯^0.5¯¯B¯¯¯^0.5¯¯¯¯¯B¯¯¯^0.5¯¯¯¯¯B¯¯¯^0.5¯¯¯¯¯B

P(All boys) = (0.5)(0.5)(0.5)(0.5)=0.0625

or all girls

Start¯¯¯^0.5¯¯G¯¯¯^0.5¯¯¯¯¯G¯¯¯^0.5¯¯¯¯¯G¯¯¯^0.5¯¯¯¯¯G

P(All girls) = (0.5)(0.5)(0.5)(0.5)=0.0625

The formula we need is the probability of the union of two mutually exclusive events:

P(A ∪ B) = P(A) + P(B)

∴ P(All boys OR All girls) = P(All boys)+ P(All girls)

= P0.0625 + 0.0625 = 0.1250

Answer 2