Let p be a probability function on a sample space S. Suppose A. b S such that P(
ID: 3134781 • Letter: L
Question
Let p be a probability function on a sample space S. Suppose A. b S such that P(A) =.36. P(A D) =.15. and P(B - A) =.48. Exercise 2. In this problem, we consider towers of lego blocks. Asmiiiic there are four colors of block available: black. white, ml and yellow. For this question, we arc interested only in the order of colors, from pottom to top. We refer to n choice as n color a rheme. For this question, consider towers of 7 blocks. How many poNslhlc color schemes are tlierc? How inauy color schemes have (he property that no two adjacent blocks have the same color? How many color scheme have no black blocks? How many color schemes have exactly three Mack blocks? How many color schemes are symmetric, in the sense that if you took the tower, and flipped it on its head, the colors would appear iu the order of the original tower?Explanation / Answer
Exercise 1:
a) P(B-A) = P(B) - P(A and B)
0.48 = P(B) - 0.15
P(B) = 0.63
b) P(A') = 1 - P(A) = 1-0.36 = 0.64
c) P(A U B) = P(A) + P(B) - P(A and B)
= 0.36 + 0.63 - 0.15 = 0.84
d) P(A' and B') = 1 - P(A U B) = 1 - 0.84 = 0.16
e) P(A' U B) = P(A') + P(B) - P(A' and B) =0.64 + 0.63 - 0.48 = 0.79
f) P(A/B) = P(A and B) / P(B) = 0.15/0.63 = 0.2381
g) P(A and B') = P(A) - P(A and B) = 0.36 - 0.0.15 = 0.21
P(A/B') = P(A and B') / P(B') = 0.21/ (1-0.63) = 0.5676
Exercise 2:
(a) The number of color possible shcemes are 7P4 = 840
(b) One adjust has one color out of 4 colors are 4P1
Another adjust has another colour out of remaining 3 colors is 3P1
The middle of 5 blocks has any colour, the number of ways 5P4
Total number of Possibilities are 4P1 * 3P1 * 5P4 = 1440
(c) Remove Black color, the remaining colors are 3. The number of ways allocate 3 colors in 7 blocks is 7P3 = 210
(d) Choose 3 blocks out 7 blocks in 7C3 ways and allocate black color, the remaining blocks 4 and allocate remining colors in 4P3 ways.
Total possibilities 7C3 * 4P3 = 840
(e) First i aranged one model as required patterns
B W R Y R W B
First 4 blocks are need for symmetric model. The number of ways 4 colors in first 4 blocks are arranged in 4P4 = 24 ways
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