Question

People at the state fair were surveyed about which type of lemonade they preferred. The results are shown below.

Pink lemonade: 156 males, 72 females
Yellow lemonade: 104 males, 48 females

The events "prefers pink lemonade” and "female” are independent because

P(pink lemonade | female) = P(pink lemonade) = 0.6.
P(female | pink lemonade ) = P(pink lemonade) = 0.3.
P(pink lemonade | female) = 0.3 and P(pink lemonade) = 0.6.
P(female | pink lemonade ) = 0.3 and P(pink lemonade) = 0.6.

Answer 1

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