Question 1. People with albinism have little pigment in their skin, hair, and eyes. The gene that governs albinism has two forms (alleles), which are denoted by a and A. Each person has a pair of these genes, one inherited by from each parent. A child inherits one of each parent's two alleles, independently with probability 0.5. Albinism is a recessive trait, so a person is albino only of the inherited pair is aa.
a) Beth's parents are not albino but she has an albino brother. What types must Beth's parent's be AA? Aa? aa?
b) Which of the types AA, Aa, aa, could a child of Beth's parents have? What is the probability for each type?
c) Beth is not albino. What are the conditional probabilities for Beth's possible genetic types?
Assume that the probabilities for Beth's genetic types are given by part (c) above. Beth marries Bob who is albino.
d) What is the conditional probability that a child of Beth and Bob is non-albino if Beth is of type Aa?
e) What is the conditional probability that a child of Beth and Bob is non-albino if Beth is of type AA?
Beth and Bob's first child is non-albino. What is the conditional probability that Beth is of type AA? Hint: Make a probability tree
Question 2:
A gambler plays a sequence of games that she either wins or loses. The outcomes of the games are independent, and the probability that the gambler wins is 2/3. The gambler stops playing as soon as she either has won a total of 2 games or has lost a total of 3 games. Let T be the number of games played by the gambler.
a) Find the probability mass function of T.
b) Find the mean of T.
c) Find the standard deviation of T.
Question 3: (The need for software is minimal)
In a marketing ploy to boost sales, a cereal manufacturer puts pictures of famous athletes in boxes of cereal. The manufacturer advertises that 40% of the boxes contain a picture of Lance Armstrong (a seven-time winner of the Tour de France, a famous and grueling bicycle race), 30% percent a picture Serena Williams (a tennis star), and 30% a picture of Lebron James (an NBA star). We are interested in the random variable X defined as the number of cereal boxes a person needs to buy in order to get the complete set of pictures?
a) Use Table B to simulate the random process of buying cereal boxes until all three pictures are collected. The resulting number of boxes is a simulated observation of X. Specifically, here is how this is done. Since the digits from 0 to 9 are equally likely, assign {0,1,2,3} to Lance Armstrong, {4,5,6} to Serena Williams, and {7,8,9} to LeBron James. Pick any line in table B and obtain the minimal sequence of single digits that will give you all three cards as represented by the assignment above. The length of the sequence is an observed value of X . For example, using line 101, the sequence 19223950340... gives us X = 7 (do you see why?)
b) Here comes the tedious part. Repeat (a) 50 times. Make sure you change location (line number) in table B each time when you are generating a new observation.I already did for you
Line 102 - 30 boxes
Line 103 - 22 boxes
Line 104 - 16 boxes
Line 105 - 19 boxes
Line 106 - 15 boxes
Line 107 - 23 boxes
Line 108 - 29 boxes
Line 109 - 39 boxes
Line 110 to 111 - 53 boxes
Line 111 - 18 boxes
Line 112 - 25 boxes
Line 113 - 22 boxes
Line 114 - 19 boxes
Line 115 - 28 boxes
Line 116 - 19 boxes
Line 117 - 23 boxes
Line 118 - 21 boxes
Line 119 to 120 - 41 boxes
Line 120 - 20 boxes
Line 121 - 29 boxes
Line 122 - 42 boxes
Line 123 - 26 boxes
Line 124 - 30 boxes
Line 125 - 34 boxes
Line 126 - 23 boxes
Line 127 - 26 boxes
Line 128 - 18 boxes
Line 129 - 29 boxes
Line 130 - 33 boxes
Line 131 - 15 boxes
Line 132 - 23 boxes
Line 133 - 26 boxes
Line 134 - 31 boxes
Line 135 - 18 boxes
Line 136 - 29 boxes
Line 137 - 25 boxes
Line 138 - 18 boxes
Line 139 - 42 boxes
Line 140 - 34 boxes
Line 141 - 19 boxes
Line 142 - 24 boxes
Line 143 - 34 boxes
Line 144 - 45 boxes
Line 145 - 31 boxes
Line 146 - 27 boxes
Line 147 - 24 boxes
Line 148 - 57 boxes
Line 149 - 17 boxes
Line 150 - 20 boxes
Line 151 - 31 boxes
c) Make a table (such as the one below) of the approximate distribution p(x) of X based on the 50 observations you obtained in (b).
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x
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3
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4
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5
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6
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7
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8
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9
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Etc...
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Count
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p(x)
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d) Obtain a histogram for the approximate distribution of X and describe the overall pattern. Hint: You can enter the 50 observations in Minitab, in C1, say, and
- Choose Graph → Histogram
- Enter C1 in the Graph variables box
- Click on the Scale button
- Select the Y-scale tab
- Choose Percent and press Enter twice.
e) Find the approximate probability that a person will have to buy 6 or less boxes to get the complete set of pictures.
f) Find the approximate mean value μ of X.