Multiplication Rule
Describe the difference between Independent and Dependent Events
Independent: One outcome has no effect on the other, event given by P(E and F) = P(E) * P(F)
Dependent: One outcome effects the outcome of another event, given by P(E and F) = P(E) * P(F|E)
What are the two formulas used in the Addition Rule? How to tell them apart?
P(E or F) = P(E) + P(F) - P(E and F) when you are asked the outcome of either E event or F event, all combined outcomes of both events.
P(E and F) = P(E) + P(F) - P(E or F) when you are asked the outcome of what the two events share. What do they have in common?
What is the formula for a binomial distribution? What does each part tell you?
n, x, p, q
P(X = x) = nCx * p^x * q^(n-x)
n= number of trials
x = number of successes
p= probability of success
q = (1-p) probability of failure
A deck of 52 cards is shuffled. If you draw a red card, what is the probability that it is a heart?
(26 red, 26 black, 13 hearts)
P(Heart|Red) = 13/26 = 1/2
A spinner has 4 sections labelled 1,2,3, and 4. If you spin it twice, what is the probability of landing on 3 both times?
(1/4) * (1/4) = 1/16 = 0.0625
In a standard deck of 52 cards, what is the probability of drawing either a king OR a queen?
(4 kings & 4 queens)
P(K or Q) = P(K) + P(Q) - P(K and Q)
(4/52) + (4/52) - 0 = 8/52 = 2/13
A six-sided die is rolled 90 times. How many times can we expect to roll a 3?
Expected Value = (Number of Trials) * (Probability of Success)
= 90 * 1/6
= 15
A school has 200 students. 120 students play soccer, and 50 of those also play basketball. If a randomly chosen student plays soccer, what is the probability they also play basketball?
50/120 = 5/12= 0.416
A bag contains 8 red marbles and 6 blue marbles. You randomly select one marble, put it back, then select another marble. What is the probability of drawing two red marbles?
(8/14) * (8/14) = 16/49 = 0.3265
In a class of 30 students, 18 students play soccer, 12 play basketball. If a student is chosen at random, what is the probability that they play both sports, given that the probability of a student playing either soccer or basketball is 23/30?
P(S and B) = P(S) + P(B) - P(S or B)
18/30 + 12/30 - 23/30 = 7/30 = 0.233
What is the probability of getting exactly 6 heads in 10 coin tosses?
x= 6, n = 10, p = 0.5
P(X = 6) = 10C6 * (0.5)^6 * (1-0.5)^(10-6)
=0.2051
In a class of 50 students, 30 study Math, and 20 study science. If 12 students study both subjects, what is the probability that a randomly selected student studies Science given that they study Math?
P(S|M) = 12/30= 0.4
A box contains 10 chocolates; 6 milk chocolate and 4 dark chocolate. You randomly select two chocolates without replacement. What is the probability that both chocolates are dark?
(4/10) * (3/9) = 2/15 = 0.133
A survey found that 60% of people like coffee, 30% like tea, and 15% like both. What is the probability that a randomly chosen person liked either coffee or tea?
P(C or T) = P(C) + P(T) - P(C and T)
0.60 + 0.30 - 0.15 = 0.75
A multiple choice quiz consists of 10 questions, each with 4 possible answers, but only one correct answer per question. Assuming a student guesses on all questions, what is the probability that the student answers exactly 6 questions correctly?
n = 10, x = 6, p = 1/4 = 0.25
P(X = 6) = 10C6 * (0.25)^6 * (1-0.25)^4=0.0162
A company has 200 employees, 120 in marketing and 124 employees have their Master's degree. Of those working in marketing, 84 have a Master's. If a randomly selected employee has a master's degree, what is the probability they work in sales?
Marketing Sales Total
Masters 84 124
No Masters
Total 120 200
Marketing Sales Total
Master's 84 40 124
No Master's 36 40 76
Total 120 80 200
40/124 = 0.322