Independent or dependent?
In a group of 20 people what is the probability the 2 people have the same birthday?
Find the probability of the opposite event
Dependent
To determine whether the events of two people having the same birthday in a group of 20 are independent or dependent, we need to understand the definition of these terms.
Independent events are events where the outcome or occurrence of one event does not affect the outcome or occurrence of another event. In other words, the probability of one event happening does not change based on whether or not another event has occurred.
Dependent events, on the other hand, are events where the outcome or occurrence of one event does affect the outcome or occurrence of another event. In this case, the probability of one event happening changes based on whether or not another event has occurred.
In the scenario you provided, we are interested in the probability of two people having the same birthday in a group of 20. Determining whether this is an independent or dependent event depends on whether the birthdays are being chosen randomly or if there are restrictions or conditions that affect the outcomes.
In this case, we assume that the birthdays are chosen randomly, which means that each person has an equal chance of being born on any specific day of the year. Under this assumption, we can consider the events of two people having the same birthday as independent.
To calculate the probability, we can use the principle of "complementary probability." We find the probability of the opposite event (no two people having the same birthday) and subtract it from 1 to get the probability of two people sharing the same birthday.
To find the probability that no two people have the same birthday, we consider the first person and his/her birthday as fixed. The second person then has 364 possible birthdays that are different from the first person's birthday. The third person has 363 possible birthdays different from the first two people, and so on. So, the probability of no two people having the same birthday can be calculated as:
(365/365) * (364/365) * (363/365) * ... * (347/365)
To find the probability that at least two people share the same birthday, we subtract the above probability from 1:
1 - [(365/365) * (364/365) * (363/365) * ... * (347/365)]
Calculating this expression will give us the probability you are looking for.