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Probability Notation and Basic Problems

1. If P(A) = 0.0084, find the value of P(A) (This is already given as 0.0084).

Algebra Class Probabilities

2. In an Algebra class, there are:

   • 32 STEM majors

   • 27 non-STEM majors

   • 12 STEM majors are sophomores; rest are freshmen

   • 17 non-STEM majors are freshmen

   • Total students = 32 + 27 = 59

   Find the following probabilities:

   a. The probability that the student is a STEM major:

   • P(STEM) = (Number of STEM majors) / (Total students) = 32 / 59 b. The probability that the student is a freshman non-STEM major:

   • P(Freshman Non-STEM) = (Number of Freshman Non-STEM) / (Total students) = 17 / 59 c. The probability that the student is a sophomore:

   • P(Sophomore) = (Number of Sophomores) / (Total students) = 12 / 59 d. The probability that the student is a freshman STEM major or a sophomore non-STEM major:

   • Freshman STEM = 32 - 12 = 20

   • Total = Freshman STEM + Sophomore Non-STEM = 20 + 27 - 17 = 30

   • P(Freshman STEM or Sophomore Non-STEM) = 30 / 59

Cookie Probability

3. In a batch of 36 cookies, 4 are not cooked all the way. If 2 cookies are selected without replacement, find the probability that both are not cooked:

   • P(both not cooked) = (4/36) * (3/35) = 12/1260 = 1/105

Construction Contract Probabilities

4. For two contracts assigned to firms X, Y, Z:

   List the sample space (all possible assignments):

   • (X, X), (X, Y), (X, Z)

   • (Y, X), (Y, Y), (Y, Z)

   • (Z, X), (Z, Y), (Z, Z)

   Probability that both contracts go to the same firm:

   • There are 3 outcomes where both contracts go to the same firm out of 9 total.

   • P(both contracts same firm) = 3/9 = 1/3

   Probability that firm Z gets at least one contract:

   • Outcomes where Z gets at least one contract: (Z,X), (Z,Y), (Z,Z), (X,Z), (Y,Z) = 5 outcomes

   • P(Z gets at least one) = 5/9

Marble Probabilities

5. A bag contains: 5 green, 4 red, 2 white, and 3 blue marbles.

   Find the following probabilities:

   a. The probability of drawing a green marble:

   • P(Green) = 5/14 b. The probability of drawing a blue or green marble:

   • P(Blue or Green) = (3 + 5)/14 = 8/14 = 4/7 c. If two marbles are drawn without replacement, the probability both are white:

   • P(both white) = (2/14) * (1/13) = 2/182 = 1/91 d. The probability that a marble drawn is neither blue nor red:

   • Remaining marbles = 5 Green + 2 White = 7; P(Not Blue or Red) = 7/14 = 1/2

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Marble Draw Probabilities

6. Continue from Page 1 e. Probability of drawing three white marbles without replacement:

   • P(3 white) = (2/14) * (1/13) * (0/12) = 0

Dice Rolling Probabilities

7. Rolling a pair of dice: a. Probability that the sum of the dice is 7:

   • Possible outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes

   • P(sum=7) = 6/36 = 1/6 b. Probability of rolling a sum of 9 or greater:

   • Outcomes for 9: (3,6), (4,5), (5,4), (6,3) = 4 outcomes

   • Outcomes for 10: (4,6), (5,5), (6,4) = 3 outcomes

   • Outcomes for 11: (5,6), (6,5) = 2 outcomes

   • Outcomes for 12: (6,6) = 1 outcome

   • Total = 10; P(sum>=9) = 10/36 = 5/18 c. Probability of one die being even and the other odd:

   • Outcomes: (1,2), (2,1), (1,4)...etc. Total = 18; P(one even, one odd) = 18/36 = 1/2

Radiologist Cases

8. Cases for radiologist excluding multiple body x-rays:

   Findings:

   a. Probability for child arm or hand x-ray = 28/Total px in table b. for adult neck or spine x-ray = 8/Total c. for adult pelvis x-ray = 12/Total d. for arm or hand OR leg or foot = Total arm or hand + Total leg or foot / Total e. for child patient = Total child cases / Total f. for adult pelvis or child neck/spine x-ray = Total combinations of above / Total g. Choosing two adult leg/foot x-rays without replacement; relevant probability = Cases of Adult Leg/Foot / Total; calculate similar for others. h. Two arm or hand x-rays out of selected cases;

   • Without replacement P(a child & adult arm/hand x-ray). i. With replacement; repeat steps from g and h, but with total cases squared.

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Grocery Store Generator Probabilities

9. Grocery store's generator: a. Probability of generator failing = P(failure) = 0.016 b. Two generators failing: P(both fail) = 0.016 * 0.016 = 0.000256

Traffic Accident Distribution

10. Table for traffic accidents and questions regarding probability distribution: a. Validity Check: sum of probabilities should equal 1 b. Find mean (E(X)) = Σ [x * P(x)] and standard deviation (σ = (X^2) - (E(X))^2) c. Use range rule for significant intervals; any x outside this is significant. d. Check if 5 accidents/day is significant based on earlier calculation. e. Probability for no more than 2 accidents is Sum of P(X=0) + P(X=1) + P(X=2). f. Probability for at least 4 accidents is P(X>=4), 1 - P(X<4).

Sleepwalking Study

11. Adults sleeping in groups of five and sleepwalking: a. Validity of probability distribution check = sum = 1. b. Expected adults sleepwalking = Mean as calculated previously.

    • Use probabilities to calculate.

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Sleepwalking Range Rule

12. Range Rule for sleepwalkers in groups of 5; decide if 3 is significantly high or not. a. Probability for more than 2 sleepwalkers (P(X>2)). b. Probability for no more than 1 sleepwalker (P(X<=1)).

Nausea in Drug Trial

13. Probability distribution of nausea: a. P(no more than 6 nausea) = sum of P(0) + P(1) + P(2)... + P(6) b. Is it an unlikely event? Compare to mean. c. Probability of half or more patients experiencing nausea. d. Expected patients experiencing nausea based on trials and distribution. e. Check significance for 8 as high or not. f. Probability of fewer than 4 experiencing nausea; calculate using distribution.

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Commercial Crimes and Burglary Stats

14. Burglary Statistics from 1980: a. Finding binomial probability of exactly 4 burglaries out of 9 tries using binomial formula; b. Calculate expected number of burglaries in 12 trials from earlier stat (0.23 * 12); mean = 2.76. c. Find probability of at least 6 burglaries from binomial probability.

Die and Probability Guesses

15. Rolling a die 600 times;

• Find binomial mean = np and standard deviation = sqrt(np(1-p)).

Store Purchases and Returns

16. Common return rate for online store:

• Estimate return rate based on 17% of sales; apply to total items.

Quiz Probability

17. Probability for quiz guessing at least 7 correct out of 10 using binomial probability.