The red face cards and the black cards numbered 2-9 are put into a bag. Four cards are drawn at random without replacement. Find the following probabilities:
a) All 4 cards are red
b) 2 cards are red and two cards are black
c) At least one of the red cards is red.
d) All four cards are black
CARDS:red=6; black=16 ... total 22
a.(6 nCr 4)/(22 nCr 4) = .002
b.(6 nCr 2)(16 nCr 2)/(22 nCr 4)= .246
c.1-(.249) = .751 ... get d first
d.(16 nCr 22)/(22 nCr 4) = .249
There are 6 red cards not 16 ( only face cards: 2 kings 2 queens and 2 jacks) and 16 black. 22 in total.
a) (6/22)*(5/21)*(4/20)*(3/19)
b) (16/22)*(15/21)*(6/20)*(5/19)
c) 1-((16/22)*(15/21)*(14/20)*(13/19))
d)(16/22)*(15/21)*(14/20)*(13/19))
a) The probability of drawing a red card on the first draw is 26/52. Since the cards are drawn without replacement, the probability of drawing a red card on the second draw is 25/51, on the third draw is 24/50, and on the fourth draw is 23/49. To find the probability of all four cards being red, we multiply these probabilities together.
P(all 4 cards are red) = (26/52) * (25/51) * (24/50) * (23/49) = 0.0784313725490196
b) The probability of drawing two red cards and two black cards can be calculated by finding the probability of drawing two red cards and multiplying it by the probability of drawing two black cards.
P(2 red and 2 black) = (26/52) * (25/51) * (26/50) * (25/49) = 0.0588235294117647
c) The probability of at least one of the red cards being drawn can be calculated by finding the complement of the probability of drawing all black cards.
P(at least one red card) = 1 - P(all black cards) = 1 - (26/52) * (25/51) * (24/50) * (23/49) = 1 - 0.0784313725490196 = 0.9215686274509804
d) The probability of drawing all black cards is the same as the probability of drawing all red cards, since the number of red and black cards in the bag is the same. This probability is 0.0784313725490196.
To find the probabilities of these events, we need to calculate the total number of possible outcomes and the number of favorable outcomes for each scenario.
Total number of possible outcomes:
The total number of cards in the bag is 26 (9 black cards + 17 red face cards). So, the total number of possible outcomes is the number of ways we can choose 4 cards out of the 26, which can be calculated using combinations. The formula for combinations is nCr, where n is the total number of items and r is the number of items chosen. In this case, n = 26 and r = 4, so the total number of possible outcomes is C(26, 4).
a) All 4 cards are red:
For all 4 cards to be red, we need to select 4 cards from the 17 red cards. So, the number of favorable outcomes is C(17, 4).
b) 2 cards are red and 2 cards are black:
To have 2 red and 2 black cards, we need to select 2 cards from the 17 red cards and 2 cards from the 9 black cards. The number of favorable outcomes is C(17, 2) * C(9, 2).
c) At least one of the red cards is red:
This event includes the previous two events. We can find the probability by calculating the complement of the event where none of the red cards are drawn, i.e., 4 black cards are drawn. So, the number of favorable outcomes is the total number of outcomes minus the number of outcomes where all 4 cards are black: C(26, 4) - C(9, 4).
d) All four cards are black:
To have all 4 cards black, we need to select 4 cards from the 9 black cards. So, the number of favorable outcomes is C(9, 4).
Now, let's calculate the probabilities using the formulas above.
a) Probability of all 4 cards being red = Number of favorable outcomes / Total number of possible outcomes
P(all red) = C(17, 4) / C(26, 4)
b) Probability of 2 red cards and 2 black cards = Number of favorable outcomes / Total number of possible outcomes
P(2 red, 2 black) = (C(17, 2) * C(9, 2)) / C(26, 4)
c) Probability of at least one red card = Number of favorable outcomes / Total number of possible outcomes
P(at least one red) = (C(26, 4) - C(9, 4)) / C(26, 4)
d) Probability of all 4 cards being black = Number of favorable outcomes / Total number of possible outcomes
P(all black) = C(9, 4) / C(26, 4)
Using these formulas, you can calculate the probabilities for each scenario.
a) There are 16 red-faced cards and 16 black.
First Card red = 16/32
Second card red = 15/31
Third Card red = 14/30
Fourth card red = 13/29
The probability of all occurring is found by multiplying the individual probabilities.
b) 16/32, 15/31, 16/30, 15/29 Again multiply.
c) Firgure probability of one red card, two red cards, three red cards, or four red cards. Remember the probability of the remaning cards being black in each case, except the last. Since you are interested in "either-or", add those four probabilities.
d) Same as a.