SelfTest 5

1
Self-Test 5
2
Problem 1 Box 1 contains 3 red and 5 white
balls, while box 2 contains 4 red and 2 white
balls. A ball is chosen at random from the first
box and placed in the second box without
observing its color. Then the ball is drawn from
the second box. Find the probability that it is
white.
3
Solution W1a white ball was chosen from the
Box 1 P(W1) 5/8R1 a red ball was
chosen from the Box 1 P(R1) 3/8W2 a
white ball was chosen from the Box 2The
probability of W2 depends on the previous
step. In fact, W2 can be presented as a union of
two mutually exclusive events depending on the
outcome of the first step
Fig.1 Boxes 1 (grey) and 2 in the initial state
Fig.2 Box 2 depending on the outcome of the first
step
P(W2) P(W1)P(W2W1) P(R1)P(W2R1) (5/8)
(3/7) (3/8)(2/7) 21/56 3/8.
4
Problem 2 A friend rolls two dice and tells you
that there is at least one 6. What is the
probability that the sum is at least 9?A at
least one 6 orange yellow P(A) 11/36. B
the sum is ay least 9 red green. AB red
P(AB) 7/36.P(BA) P(AB)/P(A) 7/11.
5
Some difficulties could be caused by the cards
problem. Problem 3 We draw 4 cards out of deck
of 52. Find the probability that all four values
are different given that (a) all four cards
belong to different suits (b) We drew two spades
and two hearts. case (a). Events SAll four
suits are different VAll values are
different.P(S) 134/C52,40.1055 P(VS)
13121110/ C52,40.0634 P(VS)
P(VS)/P(S)0.601 Case (b) B Two spades and two
hearts. P(B) C13,2C13.2/C52,4. P(SB)
C13,2C11,2/C52,4 P(VB) P(SB)/P(B)0.705
6
Problem 4 An urn contains 8 red, 7 blues and 5
green balls. You draw out two balls and they are
different color. What is the probability that the
two balls were red and blue.
BThe balls have different color A One is
red and the other is blue P(B) 1- p(Bc) 1-
P(both red) P(both blue) P(both green)1-
(8/20)(7/19)-(7/20)(6/19)-(5/20)(4/19)
131/190. P(A)(remember that the order can be
different !) (8/20)(7/19)(7/20)(8/19)
28/95. Notice now that A?B, meaning that
ABA.As a result, P(AB)P(A)/P(B) 56/131
0.428
7
5. Suppose 60 of the people subscribe to
newspaper A, 40 to newspaper B and 30 to both.
If we pick a person at random who subscribes to
at least one newspaper, what is the probability
that he subscribes to newspaper A?A
subscribing to A P (A) 0.6 B subscribing to
B. P (B) 0.4L1 AUB - at least one
newspaper.P (L1) P (A) P (B) P (AB) 0.6
0.4 - 0.3 0.7 P (AL1) P (AL1)/P (L1)
How to find P(A L1)? It is clear from the
picture that AL1A.
P(AL1) 0.6 P(AL1) 0.6/0.7 0.86
B
A
AB
Sample space
L1 A U B