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

Answer

Step-by-step explanation:

You would require the table with the number of males and females who prefer either pink or yellow lemonade at the state fair.

                     Pink Lemonade         Yellow Lemonade       TOTAL

Male                    156                                104                          260

Female                  72                               48                          120

TOTAL                 228                               152                         380

P(pink lemonade | female) = 72 / (48 + 72) = 72 / 120 = 0.6

P(pink lemonade) = (156 + 72) / (156 + 72 + 104 + 48) = 228 / 380 = 0.6

Therefore, The event "prefers pink lemonade" and "female" are independent because P(pink lemonade | female) = P(pink lemonade) = 0.6

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 [tex]P(P|F) = P(P) = 0.6[/tex]

To do this, we make the following representations:

[tex]P \to[/tex] Pink Lemonade

[tex]Y \to[/tex] Yellow Lemonade

[tex]M \to[/tex] Male

[tex]F \to[/tex] Female

The "prefer pink" event and "female" event are independent if:

[tex]P(P|F) = P(P)[/tex]

[tex]P(P|F) \to[/tex] the probability that a selected person prefers pink provided that the person is female

[tex]P(P) \to[/tex] the probability that a selected person is female

P(P| F) is calculated as:

[tex]P(P| F) = \frac{n(P\ n\ F)}{n(F)}[/tex]

i.e. the number of females who prefers pink divided by the number of females

From the table, we have:

[tex]n(P\ n\ F) = 72[/tex]

[tex]n(F) = 72 + 48 =120[/tex]

So, the probability is:

[tex]P(P| F) = \frac{72}{120}[/tex]

[tex]P(P| F) = 0.6[/tex]

[tex]P(P)[/tex] is calculated as:

[tex]P(P) = \frac{n(P)}{Total}[/tex]

i.e. the all people who prefer pink divided by total number of people

From the table, we have:

[tex]n(P) = 156 + 72 = 228[/tex]

[tex]Total = 156 + 72 + 104 + 48 = 380[/tex]

So, the probability is:

[tex]P(P) = \frac{228}{380}[/tex]

[tex]P(P) = 0.6[/tex]

Recall that:

The "prefer pink" event and "female" event are independent if:

[tex]P(P|F) = P(P)[/tex]

And we have:

[tex]P(P| F) = 0.6[/tex]

[tex]P(P) = 0.6[/tex]

Hence, the events are independent because:

[tex]P(P|F) = P(P) = 0.6[/tex]

Option (a) is correct

Read more about independent events at:

https://brainly.com/question/12543071

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