Independent Events
Part I
Definition: If A and B are independent events, then P( A | B) P( A) and P(B | A) P(B) .
Interpretation: If A and B are independent events means that whether one event has occurred or
not does not affect the probability of the other event.
Test for Independence: Two events A and B are independent if and only if
P( A B) P( A) P( B) .
Interpretation: If events A and B are independent, then P( A B) P( A) P( B) and if
P( A B) P( A) P( B) , then events A and B are independent.
Independence of More than Two Events: If E1 , E 2 , … , E n are independent events, then
P( E1
E2
...
En )
P ( E1 ) P ( E2 ) ... P ( En ) .
Do not confuse independent events with mutually exclusive events. Recall that if A and B are
mutually exclusive events, then A B
, which means that A and B cannot occur at the same
time, which in turn means P( A B) P( ) 0 .
1. In the given Venn diagram, the cardinality for the sets is given. The cardinality of a set is
the number of elements in the set.
A
20
B
30
20
30
a. Determine whether A and B are independent events.
i. P( A
B)
ii. P( A)
iii. P( B)
iv. P( A) P( B)
v. If P( A B) P( A) P( B) , then events A and B are independent. Are
events A and B independent? YES or NO
b. If A B
, then A and B are mutually exclusive. If A B
, then A and B
are not mutually exclusive. Are A and B mutually exclusive? YES or NO
Independent Events
2.
In the given Venn diagram, the cardinality for the sets is given.
A
10
B
30
20
40
a. Determine whether A and B are independent events.
i. P( A
B)
ii. P( A)
iii. P( B)
iv. P( A) P( B)
v. Are events A and B independent? YES or NO
b. Are A and B are mutually exclusive? YES or NO
3. In the given Venn diagram, the cardinality for the sets is given.
A
20
50
B
30
a. Determine whether A and B are independent events.
i. P( A
B)
ii. P( A)
iii. P( B)
iv. P( A) P( B)
v. Are events A and B independent? YES or NO
b. Are A and B mutually exclusive? YES or NO
Independent Events
4.
In the given Venn diagram, the cardinality for the sets is given.
A
0
B
50
50
a. Determine whether A and B are independent events.
i. P( A
B)
ii. P( A)
iii. P( B)
iv. P( A) P( B)
v. Are events A and B independent? YES or NO
b. Are A and B mutually exclusive? YES or NO
5. Complete the chart to see that independent events are a different concept than mutually
exclusive events.
Problem Number Independent Events
Mutually Exclusive Events
1
YES
or
NO
YES
or
NO
2
YES
or
NO
YES
or
NO
3
YES
or
NO
YES
or
NO
4
YES
or
NO
YES
or
NO
Independent Events
Part II
1. A computer keyboard (K) has a 0.5% failure rate, a computer monitor (M) has a 1%
failure rate, and a jump drive (J) has a 1.4% failure rate. If the computer keyboard,
monitor, and jump drive’s failures are independent of each other, what is the exact
probability that the keyboard, monitor and jump drive will all fail?
2. The chance of getting headache or getting dizzy from a new medication for depression is
70%, of getting a headache is 25%, and of getting dizzy is 60%. Are the events getting a
headache (H) and getting dizzy (D) independent? What two different formulas did you
need to solve this problem?