When discussing the two multiplication rules for probability we first need to start with defining
independent and dependent events. Independent events are when the outcome of one event does
not affect the probability outcome of the other event. For instance, event A is flipping a coin
an event B is rolling a die, whatever happens in event A, flipping the coin, has no effect on event B,
rolling the die. And dependent events are when the occurrence or outcome of the first event changes
the probability of the second event. For example drawing a card from a deck and not replacing it
and drawing another card. Now to the multiplication rules, and the first one, is that if 2 events are
independent, the probability of both occurring probability of A and B equals the probability
of A times the probability of B. So, flipping a coin as event A and rolling a die as event
B. And what is the probability of getting a head and a 6? Probability of A, getting a head in the
coin toss, equals 1 over 2 and the probability of B, rolling a 6 equals 1 over 6, multiply them
and we get 1 over 12, or 0.083, or 8.3% probability of this outcome. And you can see
this if you draw a tree diagram with a head and a 6 being one of 12 possible outcomes. As another
example let's say you're a stats nerd like me and you've calculated out your own win probabilities
for your favorite basketball team in their next 5 games. Game 1 you give them a 64% win
probability, game 2 is at 73%, game 3 is at 41%, game 4 is at 44% and game 5
is at 56%. Discarding injuries and load management issues, these are all independent events. So what is
the probability they win the next 5 games? We will assign games 1 through 5 as events A through E, and
using our first multiplication rule, probability of A and B and C and D and E, equals probability
of A times probability of B, times probability of C, times probability of D, times probability of E.
So we have 0.64 times 0.73, times 0.41, times 0.44, times 0.56, which equals 0.047, or a 4.7 probability
to win five in a row. Now for multiplication rule number 2, and it applies
when 2 events are dependent. The probability of A and B, equals probability of A times probability
of B given that A has already occurred, and in the last part of the equation here, this slash does not
mean divide, it means, as I just stated, probability of B given that A is already occurred. To use this
rule, let's say we want to find the probability of drawing 2 hearts from a deck without replacing
after drawing. There are 52 cards in a deck, 13 of them are hearts and we will assign drawing
a heart as the first card, as event A and drawing a heart as a second card as event B. So using our
equation or rule, the probability of A and B, equals probability of A, which in this case
is 13 over 52, times the probability of B given that A is already occurred, which is 12 over 51.
It's 12 over 51 because we're assuming that our first draw was a heart so there are only 12
hearts remaining, and 51, because we've removed a possible card from the deck with our first draw.
So doing the math, we get 156 over 2652, which equals 0.059, or a 5.9% probability of
drawing 2 hearts from a full deck. As another example let's say that in a certain zip code 68%
of homes have a Netflix subscription, and of these subscribers, 14% also have
an Amazon Prime subscription. Based on this data, what is the probability of randomly selecting a
house that has a subscription to both services? We're gonna use the formula for multiplication
rule number 2 here, the probability of A and B, equals the probability of A, times the probability
of B given that A has already occurred. In this situation we'll assign event A as a home with a
Netflix subscription, and event B is a home with an Amazon Prime subscription. We know that probability
of A is 0.68, or 68% as we were given that data, and we were actually given the second part of
the equation as well, the probability of B given that A has already occurred, means the probability
that a home has an Amazon Prime subscription, given they have a Netflix subscription, which is 0.14 or
14%. So multiplying these we get 0.095 or a probability of 9.5%, that a randomly
selected home has both Netflix and Amazon Prime. All right my friends, hopefully you got something
out of this video, I do have more videos right there for you, till next time, I am outta here.