Random Variables & Distribution Basics
What does "n" represent in the binomial distribution formula?
EcoSmart is a company that produces light bulbs, and the probability of a bulb being defective is 0.05. What is the probability that out of 100 randomly selected bulbs, at most 3 will be defective?
Probability of Sucesses= 0.05
Number of Trials= 100
Number of Successes (x)= At most 3
BCD (3, 100, 0.05)= 0.2578
The probability that at most 3 bulbs will be effective is 0.2578
The Tesla factory, on average, produces 5 defective cars per day. What is the probability that exactly 2 defective items will be produced tomorrow?
Average Rate of Sucess (mean)= 5
Random Variable (x)= 2
PoissonPDF(5,2)= 0.08422
The probability that exactly 2 defective items will be produced tomorrow is 0.08422 or 8.422%
What is the standard deviation of this normal distribution X~N(10,2^2)
The standard deviation can be obtained from the square root of the variance.
Vairance= 4
Square root (4)= 2
Standard Deviation= 2
What is the sum of probabilities of all possible outcomes in a probability distribution?
The sum of probabilities of all possible outcomes in a probability distribution must always equal 1.
What does discrete distribution mean?
Discrete distributions refer to probability distributions where the random variable can only take the value of countable numbers.
In a survey, 80% of people prefer Starbucks over Dunkin. If 10 people were selected at random, what is the probability that exactly 7 of them would prefer Starbucks.
Probability of Success= 0.80
Number of trials= 10
X-Value= 7
BPD(7, 10, 0.8)= 0.2013
The probability that exactly 7 people would prefer Starbucks is 0.2013
The number of car accidents that occur on the Pennsylvania Turnpike is generally 10 per day. What is the probability of having no accidents on a given day?
The average rate of Success (mean)= 10
Random Variable (x)= 0
PoissonPDF(10,0)= 0.00005
The probability of having no accidents on a given day is 0.00005 or 0.005%
What is the formula for calculating the z-score?
z= x - mean / standard deviation
If two events are mutually exclusive, what is the probability of both events occurring simultaneously?
Mutually exclusive events cannot occur together, so the probability of both events occurring simultaneously is 0
What does continuous distribution mean?
Continous distribution refers to a probability distribution in which the random variable can have any value in a certain interval or range.
Sai is a very good basketball player. He has a free throw success rate of 80%. If he attempts 10 free throws, what is the probability that he will make exactly 8 of them?
Success Rate= 0.80
Number of Trails= 10
Number of successes (x)= 8
BinomPDF(0.80,10,8)= 0.302
Therefore, the probability that Sai makes exactly 8 free throws is 0.302 or 30.2%
The number of customers arriving at Olive Garden follows a Poisson distribution with a mean of 4 customers per hour. What is the probability that exactly 3 customers will arrive in the next hour?
Average Rate of Sucess (mean)= 4
Random Variable (x)= 3
PoissonPDF(4,3)= 0.195
The probability that exactly 3 customers will arrive in the next hour is 0.195 or 19.5%
The height of all managers at Walmart is considered to be normally distributed, with a mean of 200cm and a standard deviation of 11cm. Determine the percentage of managers that are expected to be taller than 215cm?
Mean= 200
Standard Deviation= 11
Upper Bound= Infinity
Lower Bound= 215
NormCDF(200,11,215,100000)= 0.0863
1-0.0863=0.137
The percentage of the managers that are expected to be taller than 215cm is 13.7%
In a binomial distribution, what are the two parameters that determine the shape of the distribution?
The binomial distribution is determined by the number of trials and the probability of success in each trial
What does the expected value and variance refer to?
Expected value is essentially the mean, the average of all the possible outcomes. Variance is the measure of the spread of a specific probability distribution.
In a hypothetical scenario, if the probability of success in each trial is 0.3 and the number of trials is 10, what is the probability of getting exactly 3 successes?
The probability of success= is 0.3
Number of trials=10
Number of Successes (x)= 3
BinomPDF (0.3,10,3)= 0.267
The probability of getting exactly 3 successes is 0.267 or about 26.7%
A Walmart checkout line has an average of 6 customers per hour. What is the probability of having exactly 4 customers in the next hour?
The average rate of success (mean)= 6
Random Variable (x)= 4
PoissonPDF(6, 4)= 0.13385
The probability of having exactly 4 customers in the next hour is 0.13385 or 13.385%
Most colleges require SAT scores in order to get accepted. For Drexel University, scores on the SAT are roughly normally distributed with a mean of 1200 and a standard deviation of 100. What is the probability of an individual scoring above 1000 on the SAT?
Mean= 1200
Standard Deviation= 100
We first need to calculate the z-score
Z=(1000-1200)/100= -2
Probability(X > 1000) = 1 - normCDF(1000, infinity, 1200, 100)
1 - 0.9772 ≈ 0.0228
The probability of an individual scoring above 1000 on the SAT is approximately 0.0228 or 2.28%
What is the expected value of a discrete probability distribution?
The expected value of a discrete probability distribution is the weighted average of all possible outcomes, where the weights are the probabilities of each outcome
What is the formula for VAR(X)
VAR(X) = E(X2) - (E(X)2)
A fair coin is tossed 8 times. What is the probability of getting at least 6 heads?
To find the probability of getting at least 6 heads, we need to calculate the probabilities of getting exactly 6, 7, and 8 heads, and then add them together.
P(X ≥ 6) = P(X = 6) + P(X = 7) + P(X = 8)
Success Rate= 0.50
Number of Trials= 8
BinomPDF(0.5,8,6)= 0.109
BinomPDF(0.5,8,7)= 0.0625
BinomPDF(0.5,8,8)=0.0039
Add all the probability for final result
0.109+0.0625+0.0039= 0.1754
The probability of getting at least 6 heads is 0.1754 or 17.54%
The customer service department at Amazon receives an average of 5 calls per minute. What is the probability that at most 3 calls will arrive in a given minute?
Average Rate of Sucess (mean)= 5
Random Variable (x)= At most 3
P(X ≤ 3) = poissoncdf(5, 3) ≈ 0.2651
The probability that the customer service department will receive at most 3 calls within a given minute is about 0.265 or 26.5%
AMC Movies has done a study to see how much money their customers are willing to spend on snacks like popcorn and soda. It is found that the spending pattern is normally distributed with a mean of $20 and a standard deviation of $5. What percentage of customers will spend less than $10 on snacks?
Mean= 20
Standard Deviation= 5
X=10
Z Score= 10-20/ 5= -2
Corresponding Z-Score value based on standard normal distribution table is 0.02275
The percentage of customers that will spend less than $10 on snacks is about 2.28%
What is the formula for calculating the standard deviation of a normal distribution?
The standard deviation of a normal distribution is the square root of the variance.