Suppose we roll a pair of dice one time.
Let S be the event of getting a 7.
Let T be the event of getting either 7 or 9.
Are the events S and T dependent or independent ?
The events S and T are dependent.
One way to see this is to simply
note that whenever S occurs, T must also occur.
So S certainly does effect the probability of T.
Another way to show that the events are not independent is to show
that
P( S and T ) does not equal P( S ) * P( T ).
| P( S ) | = | 1 / 6 | |
| P( T ) | = | P( 7 or 9 ) = P( 7 ) + P( 9 ) = ( 6 / 36 ) + ( 4 / 36 ) | |
| = | ( 10 / 36 ) = 5 / 18 | ||
| Since "getting a 7" and "getting a 7 or 9" can both happen only by getting a 7, | |||
| P( S and T ) = P( S ) = 1 / 6 | |||
| P( S ) * P( T ) = ( 1 / 6 ) * ( 5 / 18 ) < P( S and T ) | |||