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Sagot :
Let's solve this step by step.
### Part (a): Probability that it is a 4
A standard deck of 52 cards contains 4 cards that are 4s (one from each suit: hearts, diamonds, clubs, and spades).
The probability of drawing a 4 is given by the number of favorable outcomes divided by the total number of possible outcomes:
[tex]\[ P(\text{4}) = \frac{\text{Number of 4s}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13} \][/tex]
Calculating this gives:
[tex]\[ P(\text{4}) = \frac{1}{13} \approx 0.07692307692307693 \][/tex]
### Part (b): Probability that it is a red card
A standard deck has 26 red cards (13 hearts and 13 diamonds).
The probability of drawing a red card is given by:
[tex]\[ P(\text{red}) = \frac{\text{Number of red cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2} \][/tex]
Calculating this gives:
[tex]\[ P(\text{red}) = \frac{1}{2} = 0.5 \][/tex]
### Part (c): Probability that it is a 4 OR a red card
To find the probability that the card drawn is either a 4 or a red card, we use the formula for the union of two events:
[tex]\[ P(\text{4 or red}) = P(\text{4}) + P(\text{red}) - P(\text{4 and red}) \][/tex]
From part (a) and part (b), we already have:
[tex]\[ P(\text{4}) = 0.07692307692307693 \][/tex]
[tex]\[ P(\text{red}) = 0.5 \][/tex]
We also need the probability that the card is both a 4 and red. There are 2 red 4s in the deck (4 of hearts and 4 of diamonds).
[tex]\[ P(\text{4 and red}) = \frac{\text{Number of red 4s}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26} \][/tex]
Calculating this gives:
[tex]\[ P(\text{4 and red}) = \frac{1}{26} = 0.038461538461538464 \][/tex]
Now we can combine these probabilities:
[tex]\[ P(\text{4 or red}) = 0.07692307692307693 + 0.5 - 0.038461538461538464 = 0.5384615384615384 \][/tex]
### Part (d): Probability that it is a 4 AND a red card
We have already calculated this in part (c). The probability that the card drawn is both a 4 and red is:
[tex]\[ P(\text{4 and red}) = \frac{2}{52} = \frac{1}{26} \][/tex]
Calculating this gives:
[tex]\[ P(\text{4 and red}) = \frac{1}{26} = 0.038461538461538464 \][/tex]
### Summary of the Results
(a) The probability that it is a 4:
[tex]\[ P(\text{4}) = 0.07692307692307693 \][/tex]
(b) The probability that it is a red card:
[tex]\[ P(\text{red}) = 0.5 \][/tex]
(c) The probability that it is a 4 OR a red card:
[tex]\[ P(\text{4 or red}) = 0.5384615384615384 \][/tex]
(d) The probability that it is a 4 AND a red card:
[tex]\[ P(\text{4 and red}) = 0.038461538461538464 \][/tex]
### Part (a): Probability that it is a 4
A standard deck of 52 cards contains 4 cards that are 4s (one from each suit: hearts, diamonds, clubs, and spades).
The probability of drawing a 4 is given by the number of favorable outcomes divided by the total number of possible outcomes:
[tex]\[ P(\text{4}) = \frac{\text{Number of 4s}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13} \][/tex]
Calculating this gives:
[tex]\[ P(\text{4}) = \frac{1}{13} \approx 0.07692307692307693 \][/tex]
### Part (b): Probability that it is a red card
A standard deck has 26 red cards (13 hearts and 13 diamonds).
The probability of drawing a red card is given by:
[tex]\[ P(\text{red}) = \frac{\text{Number of red cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2} \][/tex]
Calculating this gives:
[tex]\[ P(\text{red}) = \frac{1}{2} = 0.5 \][/tex]
### Part (c): Probability that it is a 4 OR a red card
To find the probability that the card drawn is either a 4 or a red card, we use the formula for the union of two events:
[tex]\[ P(\text{4 or red}) = P(\text{4}) + P(\text{red}) - P(\text{4 and red}) \][/tex]
From part (a) and part (b), we already have:
[tex]\[ P(\text{4}) = 0.07692307692307693 \][/tex]
[tex]\[ P(\text{red}) = 0.5 \][/tex]
We also need the probability that the card is both a 4 and red. There are 2 red 4s in the deck (4 of hearts and 4 of diamonds).
[tex]\[ P(\text{4 and red}) = \frac{\text{Number of red 4s}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26} \][/tex]
Calculating this gives:
[tex]\[ P(\text{4 and red}) = \frac{1}{26} = 0.038461538461538464 \][/tex]
Now we can combine these probabilities:
[tex]\[ P(\text{4 or red}) = 0.07692307692307693 + 0.5 - 0.038461538461538464 = 0.5384615384615384 \][/tex]
### Part (d): Probability that it is a 4 AND a red card
We have already calculated this in part (c). The probability that the card drawn is both a 4 and red is:
[tex]\[ P(\text{4 and red}) = \frac{2}{52} = \frac{1}{26} \][/tex]
Calculating this gives:
[tex]\[ P(\text{4 and red}) = \frac{1}{26} = 0.038461538461538464 \][/tex]
### Summary of the Results
(a) The probability that it is a 4:
[tex]\[ P(\text{4}) = 0.07692307692307693 \][/tex]
(b) The probability that it is a red card:
[tex]\[ P(\text{red}) = 0.5 \][/tex]
(c) The probability that it is a 4 OR a red card:
[tex]\[ P(\text{4 or red}) = 0.5384615384615384 \][/tex]
(d) The probability that it is a 4 AND a red card:
[tex]\[ P(\text{4 and red}) = 0.038461538461538464 \][/tex]
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