hould We Convict? 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 to make its case against the defendants. The probabilities below were presented by the prosecution for the known characteristics of the thieves. 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 ) = ( n C x ) ⋅ p x ⋅ ( 1 − p ) n − x , w h e r e 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.
Compound Probability
Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent.
Tree diagram
Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described.
Conditional Probability
By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.
hould We Convict?
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
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 ) = ( n C x ) ⋅ p x ⋅ ( 1 − p ) n − x , w h e r e 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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