In an experimental study, researchers had each of their participants bet on each

Question

In an experimental study, researchers had each of their participants bet on each game of a professional football season. In the contingency table below is some information from a random sample of 100 bets from this study placed on the Columbus Crush (picking them to win) during the last 14 games of the season (the Crush had 7 wins and 7 losses over that period). The table indicates, for each bet placed on the Crush, whether or not the team won and how the participant who placed the bet wagered the following week In the cells of the table are the respective observed frequencies, and three of the cells also have blanks. Fill in these blanks with the frequencies expected if the two variables, result of picking the Crush and bet placed the following week, are independent Round your responses to at least two decimal places Bet placed the following week Picked Crush to Picked Crush to win lose 43 14 Crush won 57 Result of picking the Crush 25 18 Crush lost 43 Total 68 32 100 Clear Undo Help

Explanation / Answer

We know that if 2 events A and B are independent then

P(A and B) = P(A) P(B)

Let us consider the first cell, which corresponds to A="Crush won" B=Picked Crush won

From the row totals we know that 57 of the 100 bets Crush won. That is the marginal probability of crush winning is

P(Crush won) = 57/100

From the column totals we know that 68 out of 100 times picked crush to win. That means the marginal probability of picking crush to win is

P(picked crush to win) = 68/100

We want the joint probability of

P("Crush won" and "picked crush to win") = P("Crush won") x P("picked crush to win") = (57/100)*(68/100)

There are 100 bets placed, the expected frequency of the first cell is

P("Crush won" and "picked crush to win") *100 = (57/100)*(68/100)*100 = 38.76

Similarly row 1 column 2 expected frequency is

P("Crush won" and "picked crush to lose") *100 = (57/100)*(32/100)*100 = 18.24

Row 2 column 2, the expected frequency is

P("Crush lost" and "picked crush to lose") *100 = (43/100)*(32/100)*100 = 13.76

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