Problem on probability

Question

Three cards are selected at random without replacement from a deck which contains 3 red, 3blue, 3 green, and 3 black cards. Give a sample space for this experiment and define the events:

A: all the selected cards are red

B: 1 card is red, 1 green and 1 blue

C: 3 different colors occur

D: all 4 colors occur.

Answer 1

The sample space is the list of possible outcomes for the experiment. As you are picking 3 cards each outcomes contains one card color.

Sample space will have 3x3x3x3 = 81 outcomes [ first card can be R , B, G, B .... same for the second and third card].

All the selected cards are red : R R R (3/12) X (2/11) X (1/10). [DENOMINATOR CHANGES AS IT IS WITHOUT REPLACEMENT]

Red green blue

R B G (3/12) X (3/11) X (3/10)

We multiply this probability by 6 because the r b g can occur in 6 ways

rbg rgb brg bgr grb gbr

All four colors occur.

Obviously 0 because it's impossible (we only pick 3 cards)

3 different colors occur.

We can pick the 3 different colors in 4 ways [ basically we choose a color to leave out in 4 ways]

Having fixed the 3 colors, the probability of getting those is the probability from (b)

So you need to multiply the probability from (b) by 4.