When to add and when to multiply?
Calculating probabilities for experiments involving multiple elements may be a little confusing when should you add and when should you multiply two probabilities to get a final probability
for an outcome/event? The basic rule is that if two things can occur at the same time, then you
multiply; and if they can't occur at the same time, then you add. Look at the following examples.
1. Flipping two fair coins together
Outcome
Probability
HH
½x½=¼
HT
½x½=¼
TH
½x½=¼
TT
½x½=¼
Note
Both (e.g., H and H in the first outcome)
can happen at the same time, so you
multiply
Now if we use the above experiment to find probability of x (number of heads) = 1
P(x =1) = P(HT, TH) = ¼ + ¼ = ½ . Here, you add because both HT and TH can't happen at
the same time on the same trial (only one of them will happen on a single trial).
2. Throwing a pair of fair dice
Let A = {sum of the dots on the two dice = 7} = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}
P(A) = 1/36 + 1/36 + 1/36 + 1/36 + 1/36 + 1/36 = 6/36 = 1/6. Here you add because all of the
6 outcomes can't happen together on the same trial (only one of those 6 outcomes would
occur on a single trial).
3. Picking three M&M candies
Pick 3 M&M candies from a container containing thousands of M&M's of different colors
(let's say, red colored M&M's account for 20% of the total). Our trial consists of picking 3
M&M's randomly.
P(3 red picks in a row) = P(red) x P(red) x P(red) = 0.2 x 0.2 x 0.2 = 0.008. Here you
multiply because all three can happen together during the same trial.
4. Counting the possible number of outfits
A saleswoman is taking 4 blouses and 2 skirts on her business trip. How many different
outfits can be created (assuming there is no color coordination problem)? Here you multiply
(4 x 2 = 8 different possible outfits) because each of the blouses and each of the skirts can be
worn at the same time.
Now, if she took 4 blouses, 2 skirts, and 2 pairs of pants, then what? First of all, she can use
either a skirt or a pair of pants (i.e., she can't use both of them at the same time). Therefore,
first you add 2 skirts and 2 pairs of pants to get the total number of lower wear, which would
be 2 + 2 = 4. Then using 4 blouses and 4 pieces of lower wear, you get the total number of
possible outfits as 4 x 4 = 16 (here you multiply because she can use a blouse and a lower
wear together at the same time).