Page 120 - DMTH404_STATISTICS

Statistics

Notes Alternative Method :

The above problem can also be attempted by writing various probabilities in the form of
following table :

A C Total
B
P (A  D ) P (B D ) P (C  D )
D 0.035
 0.012  0.015  0.008
P
P
P (A  D ) (B D ) (C  D )
D 0.965
 0.588  0.285  0.092
0.600 0.300 0.100 1.000
Total
0.012
)
Thus ( /P A D  etc.
0.035
Example 55: A box contains 4 identical dice out of which three are fair and the fourth is
loaded in such a way that the face marked as 5 appears in 60% of the tosses. A die is selected at
random from the box and tossed. If it shows 5, what is the probability that it was a loaded die?

Solution.
Let A be the event that a fair die is selected and B be the event that the loaded die is selected from
the box.

3 1
P
P A
Then, we have ( )  and ( ) B  .
4 4
Further, let D be the event that 5 is obtained on the die, then
a 1
P D/Af = and ( /P D B  6
)
6 10
3 1 1 6 11
Thus, P(D) = P(A).P(D/A) + P(B).P(D/B)  ´ + ´ 
4 6 4 10 40
We want to find P(B/D), which is given by

P (B  D ) 1 6 40 6
)
P ( /B D   ´ ´ 
( )
P D 4 10 11 11
Example 56: A bag contains 6 red and 4 white balls. Another bag contains 3 red and 5
white balls. A fair die is tossed for the selection of bag. If the die shows 1 or 2, the first bag is
selected otherwise the second bag is selected. A ball is drawn from the selected bag and is found
to be red. What is the probability that the first bag was selected?

Solution.
Let A be the event that first bag is selected, B be the event that second bag is selected and D be the
event of drawing a red ball.

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