Question

State the number of friends (or connections) that you have on Facebook (or Linkedin). In case you have more than 365 friends or connections, think of an alternative, smaller group of friends or relatives. What is the chance that there are at least 2 people among your friends (or connections) with the same birthday (same day, not same year)? Let's find out. Please respond with an estimate of the probability that this will happen. This estimate can be intuitive or you can do some calculations: both ways are ok. In case you know the birthdays of your friends (or connections), check if your estimate of the probability corresponds to the reality.

A street performer approaches you to make a bet. He shows you three cards: one that is blue on both sides, one that is orange on both sides, and one that is blue on one side and orange on the other. He puts the cards in the bag, pulls out one, and puts it on the table. Both of you can see that the card is blue on top, but haven't seen the other side. The street performer bets you $50 that the other side of the card is also blue. Should you take the bet and WHY?

Now that the previous two questions have gotten you thinking about probability, how does probability apply to your (desired) profession?

Please note that it is extremely important that you answer the above questions by yourself, without consulting your classmates. If I notice any similarity of postings, none of the students involved will get any credit!

Answer 1

Concept: Probability is basically how likely something is going to happen.

Probability = number of ways an event can happen/ Total number of possible outcomes. For example, if we flip a coin, there are two possibilities {H, T} Now, if we want to calculate the probability of a head = 1/2

Solution

I have more than 365 friends on Facebook/LinkedIn. So, I am choosing a group of my college friends group which has around 53 people. I just think that (without any calculation), that at least two people having the same birthday would be 80%.

Now if I try calculating, it would be as follows

Lets say first person's birthday is on any day,

So, the probability that 2nd person's birthday would be on the same day is 1/365 as there are 365 options and only 1 favorable outcome. Now, we need to find the probability for a group of 53 people.

probability at least 2 people share the same birthday = 1 - the probability that NO people share the same birthday

Let's calculate the probability that NO people share the same birthday

• As we discussed, let's say that first-person has a birthday on any day, so the probability is 365/365
• Now the 2nd person has only 364 option, so the probability is 364/365 as we don't want same birthday
• Similarly, 3rd person has a probability 363/365 and so on till 53rd person

So the probability that NO people share the same birthday = 365/365 * 364/365* 363/365*............312/365

= = 0.0188

ie 1.88 % that no people would share the same birthday

So, the probability that at least two-person would have same birthday is 1-.0188 = 0.9812 = 98.12%, which says that there is very high chance that in my group there would be at least two-person sharing the same birthday

In actual, there are 2 pairs who have the same birth dates

Part B

Since the card on the table has one side blue in it. Now there are only two options for that card. Either it is the one which has both sides blue or it is the one with orange on one side. So, the probability of another side being also blue is 1/2 meaning there is a 50% chance of winning the bet. If we calculate the expected value = 0.5*50 +0.5*(-50) = 0. So I won't take the bet.

Probability is applied in our daily activities of work. We, plan our customer's reaction with some probability, price our product keeping in mind the customer reaction (based on probability) and so on. We also do our hypothesis testing for our software bugs. So, it is heavily involved