Consider the experiment of drawing two cards from a deck in which all pictures cards have been removed and adding their values (with ace = 1).
A. Describe the outcomes of this experiment. List the elements of the sample space.
B. What is the probability of obtaining a total of 5 for the two cards?
C. Let A be the event "total card value is 5 or less. "Find P (A) and P (Ac).
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A. In this experiment, we are drawing two cards from a standard deck in which all picture cards (i.e., king, queen, and jack) have been removed. The deck now contains only numbered cards (2-10) and Aces. The elements of the sample space can be listed as follows:
1 + 1 = 2
1 + 2 = 3
1 + 3 = 4
1 + 4 = 5
1 + 5 = 6
1 + 6 = 7
1 + 7 = 8
1 + 8 = 9
1 + 9 = 10
1 + 10 = 11
2 + 1 = 3
2 + 2 = 4
2 + 3 = 5
2 + 4 = 6
2 + 5 = 7
2 + 6 = 8
2 + 7 = 9
2 + 8 = 10
2 + 9 = 11
3 + 1 = 4
3 + 2 = 5
3 + 3 = 6
3 + 4 = 7
3 + 5 = 8
3 + 6 = 9
3 + 7 = 10
3 + 8 = 11
4 + 1 = 5
4 + 2 = 6
4 + 3 = 7
4 + 4 = 8
4 + 5 = 9
4 + 6 = 10
5 + 1 = 6
5 + 2 = 7
5 + 3 = 8
5 + 4 = 9
6 + 1 = 7
6 + 2 = 8
6 + 3 = 9
7 + 1 = 8
7 + 2 = 9
8 + 1 = 9
B. To find the probability of obtaining a total of 5 for the two cards, we need to determine the number of favorable outcomes (sum of two cards equals 5) and divide it by the total number of possible outcomes.
The favorable outcomes are: (1, 4), (2, 3), (3, 2), and (4, 1). So, there are 4 favorable outcomes.
The total number of possible outcomes is the number of elements in the sample space, which is 36.
Therefore, the probability of obtaining a total of 5 for the two cards is 4/36, which simplifies to 1/9.
C. Let A be the event "total card value is 5 or less."
To find P(A), we need to determine the number of favorable outcomes (sum of two cards is 5 or less) and divide it by the total number of possible outcomes.
The favorable outcomes for event A are: (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1). So, there are 9 favorable outcomes.
The total number of possible outcomes remains 36.
Therefore, the probability of event A, P(A), is 9/36, which simplifies to 1/4.
To find P(Ac), the probability of the complement of event A (i.e., total card value is not 5 or less), you can subtract P(A) from 1.
P(Ac) = 1 - P(A) = 1 - 1/4 = 3/4.