Why does this probability scenario hurt everyone's brain so much, asks maths lecturer Dr John Moriarty. Imagine Deal or No Deal with only three sealed red boxes.
You pick a box, let's say box two, and the dreaded telephone rings. The Banker tempts you with an offer but this one is unusual. Each year at my university we hold open days for hordes of keen A-level students. We want to sell them a place on our mathematics degree, and I unashamedly have an ulterior motive - to excite the best students about probability using this problem, usually referred to as the Monty Hall Problem.
People just can't seem to wrap their heads around it. This mind-melter was alluded to in an AL Kennedy piece on change this week and dates back to Steve Selvin in when it was published in the academic journal American Statistician.
It imagines a TV game show not unlike Deal or No Deal in which you choose one of three closed doors and win whatever is behind it. One door conceals a Cadillac - behind the other two doors are goats. The game show host, Monty Hall of Let's Make a Deal fame , knows where the Cadillac is and opens one of the doors that you did not choose.
You are duly greeted by a goat, and then offered the chance to switch your choice to the other remaining door. In these cases, very large sample sizes would be needed in order to estimate such probabilities to a good standard of relative accuracy. Here statistical models can help, depending on the context. For example, consider estimating the probability that the lowest of the maximum daily temperatures at a site in February in any one year is less than zero degrees Celsius.
A record of such temperatures in past years could be used to estimate this probability. A model-based alternative would be to select of family of probability distributions and fit it to the data set containing the values of years past. The fitted distribution would provide an alternative estimate of the desired probability. This alternative method can provide an estimate of the probability even if all values in the record are greater than zero. Privacy Policy.
Skip to main content. Combinatorics and Probability. Search for:. Learning Objectives Explain the most basic and most important rules in determining the probability of an event. Key Takeaways Key Points Probability is a number that can be assigned to outcomes and events. It always is greater than or equal to zero, and less than or equal to one. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.
Key Terms event : A subset of the sample space. Learning Objectives Give examples of the intersection and the union of two or more sets. Key Takeaways Key Points The union of two or more sets is the set that contains all the elements of the two or more sets. The intersection of two or more sets is the set of elements that are common to every set.
Key Terms independent : Not contingent or dependent on something else. Key Terms conditional probability : The probability that an event will take place given the restrictive assumption that another event has taken place, or that a combination of other events has taken place independent : Not dependent; not contingent or depending on something else; free.
Learning Objectives Explain an example of a complementary event. An event and its complement are mutually exclusive, meaning that if one of the two events occurs, the other event cannot occur.
An event and its complement are exhaustive, meaning that both events cover all possibilities. Key Terms mutually exclusive : describing multiple events or states of being such that the occurrence of any one implies the non-occurrence of all the others exhaustive : including every possible element. Learning Objectives Calculate the probability of an event using the addition rule. Key Terms sample space : The set of all possible outcomes of a game, experiment or other situation.
Learning Objectives Explain the concept of independence in relation to probability theory. Key Terms independence : The occurrence of one event does not affect the probability of the occurrence of another. Learning Objectives Calculate the empirical probability of an event based on given information.
Out of 2 Boys and 2 Girls, two students are chosen to advance to the next level. What is the probability that two girls advance to the next level. However, because the question was ambiguous I calculated the probabilities considering all four cases of whether they were distinguishable or indistinguishable, and whether order mattered or didn't matter and got different probabilities for each case.
However, I want to know why this happens. All you are doing is simply picking two students and seeing if they are both girls. You keep doing this and after infinite trials, divide the number of times they were both girls by the number of total trials. How does this outcome depend upon whether you view them as distinguishable, indistinguishable, ordered, or non-ordered? The difference comes about due to ambiguity of the instruction, "choose a student.
Your calculations are done under a different interpretation: that you are sampling uniformly from the set of however you're representing the outcomes. For a clearer example of the distinction, suppose you had a classroom of 1 million girls and one single boy. Now pick two students; what is the probability that you have one of each sex? If you uniformly pick from the set of pairs of students, the probability is overwhelming that you will get two girls.
Usually when we say "pick randomly" we mean the former, but like you say the latter answer is not a "wrong" interpretation per se. EDIT: A bit more explanation that might be helpful. What's the probability of rolling a 2 on a loaded die? It all depends on how the die is loaded. Different ways of loading the die will give you different results, and you will need to modify the formula above to correctly weight the different outcomes based on exactly how the die is loaded. Now, to your problem of picking students.
The usual interpretation of the problem you quoted is that you want to pick two students so that each pair of students is equally likely to get picked. Whether or not the pairs are ordered turns out not to matter, but for now let's say the pairs are ordered.
Procedure Calculate the theoretical probability for a coin to land on heads or tails, respectively. Write the probabilities in fraction form. What is the theoretical probability for each side? Now get ready to toss your coin. Out of the 10 tosses, how often do you expect to get heads or tails?
Toss the coin 10 times. After each toss, record if you got heads or tails in your tally sheet. Count how often you got heads and how often you got tails. Write your results in fraction form. The denominator will always be the number of times you toss the coin, and the numerator will be the outcome you are measuring, such as the number of times the coin lands on tails.
You could also express the same results looking at heads landings for the same 10 tosses. Do your results match your expectations? Do another 10 coin tosses. Do you expect the same results? Why or why not? Compare your results from the second round with the ones from the first round. Are they the same? Continue tossing the coin. This time toss it 30 times in a row. Record your results for each toss in your tally sheet. What results do you expect this time?
Look at your results from the 30 coin tosses and convert them into fraction form. How are they different from your previous results for the 10 coin tosses? Count how many heads and tails you got for your total coin tosses so far, which should be Again, write your results in fraction form with the number of tosses as the denominator 50 and the result you are tallying as the numerator. Does your experimental probability match your theoretical probability from the first step?
And after you multiply your numerator by 2, you will have a number that is out of —and a percentage.
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