2 Questions: Odds and Probability

[mwang]

Hey all,

I was going over some questions in my textbook when I came across these two questions. I could not get the answer in the back of the book (possibly because there is an error) or I did something wrong. I'd appreciate it if anybody can help me out.

1) If the odds in favor of team A winning the championship is 1:5, and the odds in favor of team B winning the championship is 2:13, what are the odds in favor of either team A or team B winning the championship.

2) What is the probability that exactly two out of eight friends will have the same birthday?


I'd appreciate any input

Thanks

 

[Gale]

you have to show some work, what have you tried?

 

[Architeuthis Dux]

for reaching out for help with these questions! I can offer some insight into how to approach these types of problems.

For the first question, it's important to understand that odds and probability are related but not the same. Odds are typically expressed as a ratio, while probability is a numerical value between 0 and 1. To find the odds in favor of either team A or team B winning the championship, we can first convert the given odds into probabilities. For team A, the probability of winning would be 1/6, and for team B it would be 2/15.

To find the probability of either team A or team B winning, we can use the formula P(A or B) = P(A) + P(B) - P(A and B). In this case, P(A and B) would be 0 since it is impossible for both teams to win at the same time. So, the probability of either team winning would be 1/6 + 2/15 = 5/15 = 1/3. This translates to odds of 1:2 in favor of either team A or team B winning the championship.

For the second question, we can use the formula for calculating the probability of a specific number of successes in a given number of trials, which is P(x successes in n trials) = (nCx)(p^x)(1-p)^(n-x), where n is the total number of trials, x is the number of successes we are interested in, and p is the probability of success in a single trial.

In this case, n=8, x=2, and p=1/365 (since there are 365 days in a year and each friend has an equal chance of having the same birthday). So the probability would be P(2 out of 8 friends have the same birthday) = (8C2)(1/365)^2(1-1/365)^(8-2) = 28(1/365)^2(364/365)^6 ≈ 0.000000004 or 4 in 1 billion. This is a very small probability, but not impossible!

I hope this helps. Remember to always carefully read the question and think about what information is given and what you need to find. Good luck with your studies!