Question

A randomized controlled experiment has 50 participants, of whom 20 are women. A simple random sample of 25 participants are assigned to the treatment group and the remainder to the control group. a) Say whether the following statement is true or false. If it is true, provide a math expression for the chance. If it is false, provide math expressions for the two chances. ?(the treatment group has 16 women) = ?(the control group has 16 women) b) Say whether the following statement is true or false, and justify your answer. The event "the treatment group has 16 women" is independent of the event "the control group has 16 women". c) Write a math expression for the chance that there are at least six women in both groups. [Hint: This can be done by just thinking about the women in one of the groups. Any other way will prove quite a bit harder.]

Answer 1

a) P( the treatment group has 16 women ) =P( the control group has 16 women) is TRUE

As the participants are randomly assigned to two groups, probability that one group has 16 women is same as the other.

Mathematical expression for this probability is ²⁰C₁₆ *³⁰C_25-16 / ⁵⁰C₂₅

(And the probability is ²⁰C₁₆ *³⁰C₉ / ⁵⁰C₂₅ = 0.00055 )

Note : Out of 50 persons 25 can be chosen in ⁵⁰C₂₅ ways ( total number of cases)

16 women and 25-16  men can be chosen from 20 women and 30 men in  ²⁰C₁₆ *³⁰C₉ ways ( favorable number of cases)

Probability of an event = favorable number of cases / total number of cases.

(b) No two events are not independent

As 20 women are divided into two groups , what number one group gets influences other . If one group gets 16 women , other must have 20-16 =4 women .

(c) If we consider only one group , the minimum number of women in the group is 6 and maximum number of women is 14 (as the other group must have 6 women at least)

Thus we have to find the probability that one group has 6 to 14 women

Mathematical expression is   

where x is the number of women in one group

(Required probability = ²⁰C₆ *³⁰C₁₉ / ⁵⁰C₂₅ + ²⁰C₇ *³⁰C₁₈ / ⁵⁰C₂₅ + ........+²⁰C₁₄ *³⁰C₁₁ / ⁵⁰C₂₅

  = 0.9914 )