Question

(50 POINTS) Dylan is working on a magic trick with a deck of cards. He pulled out and replaced one card 80 times and recorded the suit of the card each time. Complete the table below.

0.275 16 20 18 0.25 0.3 0.225

Answer 1

Hearts Frequency = 16

Hearts Experimental Probability = 0.2

Clubs Frequency = 40 - y

Clubs Experimental Probability = (40 - y) / 80

Diamonds Frequency = 24

Diamonds Experimental Probability = 24 / 80 = 0.3

Spades Frequency = 20 + y

Spades Experimental Probability = (20 + y) / 80

Explanation:

To fill in the table, we need to use the formula for experimental probability:

Experimental Probability = Number of times the event occurred / Total number of trials

Hearts Frequency: We don't have this information yet.

Hearts Experimental Probability = Number of times Hearts occurred / Total number of trials = 0.2 (given)

Clubs Frequency: Dylan recorded the suit of the card 80 times, but we don't know how many times Clubs occurred. We can calculate this by subtracting the number of times Hearts, Diamonds, and Spades occurred from the total number of trials:

Clubs Frequency = Total number of trials - (Hearts Frequency + Diamonds Frequency + Spades Frequency) = 80 - (x + 24 + y)

where x is the Hearts Frequency and y is the Spades Frequency.

Clubs Experimental Probability = Clubs Frequency / Total number of trials

Diamonds Frequency: We know that Dylan recorded Diamonds 24 times.

Diamonds Experimental Probability = Diamonds Frequency / Total number of trials

Spades Frequency: We don't have this information yet.

Spades Experimental Probability = Spades Frequency / Total number of trials

To fill in the table, we need to find the values of Hearts Frequency, Clubs Frequency, and Spades Frequency:

Hearts Frequency + Diamonds Frequency + Clubs Frequency + Spades Frequency = Total number of trials = 80

We know that Hearts Experimental Probability is 0.2, so Hearts Frequency must be:

Hearts Frequency = Hearts Experimental Probability * Total number of trials = 0.2 * 80 = 16

Using this value, we can find Clubs Frequency:

Clubs frequency = Total number of trials - (Hearts Frequency + Diamonds Frequency + Spades Frequency) = 80 - (16 + 24 + y) = 40 - y

Now we have two equations for Clubs Frequency and Spades Frequency:

Clubs Frequency = 40 - y

Spades Frequency = 80 - (Hearts Frequency + Diamonds Frequency + Clubs Frequency) = 80 - (16 + 24 + (40 - y)) = 20 + y

We can substitute these equations into the formula for Clubs Experimental Probability and Spades Experimental Probability:

Clubs Experimental Probability = Clubs Frequency / Total number of trials = (40 - y) / 80

Spades Experimental Probability = Spades Frequency / Total number of trials = (20 + y) / 80

Now we can fill in the table:

Hearts Frequency = 16

Hearts Experimental Probability = 0.2

Clubs Frequency = 40 - y

Clubs Experimental Probability = (40 - y) / 80

Diamonds Frequency = 24

Diamonds Experimental Probability = 24 / 80 = 0.3

Spades Frequency = 20 + y

Spades Experimental Probability = (20 + y) / 80

Note that we don't have enough information to determine the value of y or the exact frequencies of Clubs and Spades. We can only express them in terms of y.