1.3. Dependent Events and Sample Spaces http://www.ck12.org
Example A
What’s the probability of drawing two sevens from a standard deck of cards if once 1 card is chosen it is not replaced?
In this case, the probability of drawing a seven on the second draw is dependent on drawing a seven on the first
draw. Now let’s calculate the probability of the 2 cards being drawnwithout replacement. This can be done with the
Multiplication Rule.
LetA= 1 stseven chosen.
LetB= 2 ndseven chosen.
4 suits 1 seven per suit
↘ ↙
The total number of sevens in the deck= 4 × 1 = 4.P(A) =
4
52
Note: The total number of cards isP(B) =3
51
↙51 after choosing the first card if
it is not replaced.P(AandB) =4
52
×
3
51
orP(A∩B) =4
52
×
3
51
P(A∩B) =
12
2652
P(A∩B) =
1
221
Notice in this example that the numerator and denominator decreased fromP(A)toP(B). Once we picked the first
card, the number of cards available from the deck dropped from 52 to 51. The number of sevens also decreased from
4 to 3. Again, the explanation given in the example was that the first card chosen was kept in your hand and not
replaced into the deck before the second card was chosen.
Example B
A box contains 5 red marbles and 5 purple marbles. What is the probability of drawing 2 purple marbles and 1 red
marble in successionwithout replacement?
On the first draw, the probability of drawing a purple marble is: