Answer:
Independents events are events that can happen independently on their own. Events A and B are independent if P(A|B) = P(A).
The "prefer pink" event and "female" event are independent because P(P|F) = P(P) = 0.6P(P∣F)=P(P)=0.6
To do this, we make the following representations:
P \toP→ Pink Lemonade
Y \toY→ Yellow Lemonade
M \toM→ Male
F \toF→ Female
The "prefer pink" event and "female" event are independent if:
P(P|F) = P(P)P(P∣F)=P(P)
P(P|F) \toP(P∣F)→ the probability that a selected person prefers pink provided that the person is female
P(P) \toP(P)→ the probability that a selected person is female
P(P| F) is calculated as:
P(P| F) = \frac{n(P\ n\ F)}{n(F)}P(P∣F)=
n(F)
n(P n F)
i.e. the number of females who prefers pink divided by the number of females
From the table, we have:
n(P\ n\ F) = 72n(P n F)=72
n(F) = 72 + 48 =120n(F)=72+48=120
So, the probability is:
P(P| F) = \frac{72}{120}P(P∣F)=
120
72
P(P| F) = 0.6P(P∣F)=0.6
P(P)P(P) is calculated as:
P(P) = \frac{n(P)}{Total}P(P)=
Total
n(P)
i.e. the all people who prefer pink divided by total number of people
From the table, we have:
n(P) = 156 + 72 = 228n(P)=156+72=228
Total = 156 + 72 + 104 + 48 = 380Total=156+72+104+48=380
So, the probability is:
P(P) = \frac{228}{380}P(P)=
380
228
P(P) = 0.6P(P)=0.6
Recall that:
The "prefer pink" event and "female" event are independent if:
P(P|F) = P(P)P(P∣F)=P(P)
And we have:
P(P| F) = 0.6P(P∣F)=0.6
P(P) = 0.6P(P)=0.6
Hence, the events are independent because:
P(P|F) = P(P) = 0.6P(P∣F)=P(P)=0.6
Option (a) is correct