If two cards
are drawn without replacement from a deck of cards, what is the joint
probability that both cards are red?
The multiplication
rule of probability of two dependent events is the product of the probability
of first independent marginal event and the conditional probability of second dependent
marginal event. Same applies to the joint probability of multiple dependent
events.
Symbolically,
let A be an independent event and B be a dependent event, the joint probability
that both events and A and B occur is stated as:
P(A and
B) = P(A)*P (B/A), where P(A) is the probability of an independent event A and
P(B/A) is the conditional probability of a dependent event B given the
occurrence of an independent event A.
The
events are said to be dependent events if the occurrence
of the first event affects the probability of the second and following events.
In this example, drawing of a red card in the first event and not replacing
back that card to the deck of cards before second draw of a card affects the second
draw of a card. This applies to multiple draws of cards without replacement
from a deck of cards.
Let R1
be an independent event of drawing a red card in the first draw of a card. There are 26 red cards in a deck of cards. The
marginal probability of R1, P(R1) is 26 divided by 52, 0.5. Let R2 be a dependent
event of drawing a red card from the deck of cards in which the first drawn
card is not replaced back. There are 25
red cards left in the deck of 51 cards, as the first drawn cards is red which
is not replaced back in the deck. Thus, the conditional probability of R2 given
R1 in the first draw, P(R2/R1) is 25 divided by 51, 0.4902.
Now,
applying the multiplication rule of dependent events, the joint probability that
red cards are drawn in both draws without replacement is the product of P(R1)
in the first draw and the P(R2) in the second draw.
Symbolically,
P(R1 and R2) = P(R1)*P (R2/R1)=26/52 *25/51 =0.2450
The total number of outcomes in two draws of cards without replacement
from a deck of cards and their probabilities are shown using a tree diagram below:
Figure 1:
Marginal probabilities of first draws and conditional probabilities of second draws
given the first draws of cards without replacement and joint probabilities of
possible outcomes
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