these questions are midterm example and I have trouble with 2 and 5 how can I solve ?
1. Assuming each day you drive to work a traffic light that you encounter is
either green with probability 3/8, red with probability ½, or yellow with
probability 1/8, independent of the status of the light on any other day.
If over the course of 4 days, G, Y, and R denote the number of times the
light is found to be green, yellow, or red, respectively. a) What is the
probability that P[G=1, Y=2, R=1]? b) What is the probability P[G=Y]?
2. Consider a binary code with 7 bits (0 o 1) in each code word. An example
of a code word is 0101011. In each code word, a bit is a zero with
probability 0.7, independent of any other bit. a) What is the probability
of the code word 0011011 ? b) What is the probability that a code word
contains exactly two ones?
3. An urn contains 4 black and 6 white balls. Person A and person B withdraw
balls from the urn consecutively and without replacement of the balls
drawn until a black ball is selected. Find the probability that person A is
the one who selects the black ball. (Note: person A draws the first ball,
then person B, then again A, and so on.)
4. Six people, designated as A, B, C, D, E, and F, are arranged in linear order.
Assuming each possible order is equally likely, find the probability of the
following events:
a) there is exactly one person between A and B,
b) there are exactly two people between A and B,
c) there are four people between A and B.
5. A fair coin is tossed three times and the random variable X equals the
total number of heads. Find and sketch the cumulative distribution and
probability mass function of X.
1. Assuming each day you drive to work a traffic light that you encounter is
either green with probability 3/8, red with probability ½, or yellow with
probability 1/8, independent of the status of the light on any other day.
If over the course of 4 days, G, Y, and R denote the number of times the
light is found to be green, yellow, or red, respectively. a) What is the
probability that P[G=1, Y=2, R=1]? b) What is the probability P[G=Y]?
2. Consider a binary code with 7 bits (0 o 1) in each code word. An example
of a code word is 0101011. In each code word, a bit is a zero with
probability 0.7, independent of any other bit. a) What is the probability
of the code word 0011011 ? b) What is the probability that a code word
contains exactly two ones?
3. An urn contains 4 black and 6 white balls. Person A and person B withdraw
balls from the urn consecutively and without replacement of the balls
drawn until a black ball is selected. Find the probability that person A is
the one who selects the black ball. (Note: person A draws the first ball,
then person B, then again A, and so on.)
4. Six people, designated as A, B, C, D, E, and F, are arranged in linear order.
Assuming each possible order is equally likely, find the probability of the
following events:
a) there is exactly one person between A and B,
b) there are exactly two people between A and B,
c) there are four people between A and B.
5. A fair coin is tossed three times and the random variable X equals the
total number of heads. Find and sketch the cumulative distribution and
probability mass function of X.
