Dependent and Independent Events in Probability
In this class, We discuss Dependent and Independent Events in Probability.
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Dependent Events:
Two events A and B, are said to be dependent if occurring of event A will affect the probability of event B.
Example:
A bag contains five black and six green balls.
Randomly pick a ball fom the bag two times.
A = First time, should pick a black ball
B = Second time, should pick a black ball
Probability to pick a black ball the first time = 5c1/11c1
The probability of selecting a black ball a second time depends on the first-time selection.
First attempt if we select green ball. Probability of selecting black ball second time = 5c1/10c1
First attempt if we select black ball. Probability of selecting black ball second time = 4c1/10c1
Our conditional probability class discussed how to find the probability of dependent events?
After selecting a ball the first time in the above random experiment, we are not replacing the ball in the bag.
We are selecting with replacement.
Independent Events:
First time probability of picking black ball = 5c1/11c1
Now we replace the ball in the bag.
Second-time probability to pick a black ball = 5c1/11c1
Whatever ball we pick the first time, with replacement, the probability of a second attempt is not getting affected.
We call these types of events independent events.
Independent Events:
Two events, A and B, are said to be independent events.
The occurrence of one event will not affect the probability of another event.
Another Example:
Toss a coin two times.
A = First time should toss head.
B = Second time should toss head.
When we toss the first-time coin probability of getting a head = 1/2
When we toss the coin a second time. probability of getting a head = 1/2
The first event will not affect the probability of the second event.
Independent Events:
P(A ∩ B) = P(A) P(B)
P(A ∩ B) = occurring of both the events.
From the random experiment, toss a coin two times. P(A ∩ B) = 1/2 * 1/2 = 1/4
Proof:
Toss the coin two times.
P(Both Heads)?
Sample Space S = {HH, HT, TH, TT}
Event E = {HH}
P(E) = 1/4.