woman who was shopping in Los Angeles had her purse stolen by a young, blonde female who was wearing a ponytail. The blonde female got into a yellow car that was driven by a white male who had a mustache and a beard. The police located the blonde female named Janet Collins who wore her hair in a ponytail and had a friend who was a white male who had a mustache and beard and also drove a yellow car. The police arrested the two subjects. Because there were no eyewitnesses and no real evidence, the prosecution used probability to make its case against the defendants. The probabilities below were presented by the prosecution for the known characteristics of the thieves. Probability Distribution Characteristic Probability Yellow Car 1/10 Man with Mustache 1/4 Woman with a Ponytail 1/10 Woman with Blonde Hair 1/3 White Man with Beard 3/10 Man & Woman in car 1/100 Assuming that the characteristics listed are independent of each other, what is the probability that a randomly selected couple has all these characteristics? Tha is, what is P("yellow car" and "man with mustache" and "woman with ponytail" and "woman with blonde hair" and "white man with beard" and "man/woman in a car")? Would you convict the defendants based on this probability? Now let n represent the number of couples in the Los Angeles area who could have committed the crime. Let p represent the probability that a randomly selected couple has all six characteristics listed. Let the random variable X represent the number of couples who have all the characteristics listed in the table. Assuming that the random variable X follows the binomial probability function, we have P(x) = (nCx) * px * (1 - p)n-x, where x = 0, 1, 2,..., n Assuming that there are n=1,000,000 couples in the Los Angeles area, what is the probability that more than one of them has the characteristics listed in the table (hint: 1-P(x=0)-P(x=1) by hand or in statcrunch binomial distribution calculator). Does this result cause you to change your mind regarding the defendants' guilt.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
A woman who was shopping in Los Angeles had her purse stolen by a young, blonde female who was wearing a ponytail. The blonde female got into a yellow car that was driven by a white male who had a mustache and a beard. The police located the blonde female named Janet Collins who wore her hair in a ponytail and had a friend who was a white male who had a mustache and beard and also drove a yellow car. The police arrested the two subjects.
Because there were no eyewitnesses and no real evidence, the prosecution used
Probability Distribution
| Characteristic | Probability |
|---|---|
| Yellow Car | 1/10 |
| Man with Mustache | 1/4 |
| Woman with a Ponytail | 1/10 |
| Woman with Blonde Hair | 1/3 |
| White Man with Beard | 3/10 |
| Man & Woman in car | 1/100 |
- Assuming that the characteristics listed are independent of each other, what is the probability that a randomly selected couple has all these characteristics? Tha is, what is P("yellow car" and "man with mustache" and "woman with ponytail" and "woman with blonde hair" and "white man with beard" and "man/woman in a car")?
- Would you convict the defendants based on this probability?
- Now let n represent the number of couples in the Los Angeles area who could have committed the crime. Let p represent the probability that a randomly selected couple has all six characteristics listed. Let the random variable X represent the number of couples who have all the characteristics listed in the table. Assuming that the random variable X follows the binomial probability
function , we have P(x) = (nCx) * px * (1 - p)n-x, where x = 0, 1, 2,..., n Assuming that there are n=1,000,000 couples in the Los Angeles area, what is the probability that more than one of them has the characteristics listed in the table (hint: 1-P(x=0)-P(x=1) by hand or in statcrunch binomial distribution calculator). Does this result cause you to change your mind regarding the defendants' guilt.
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