Calculating Probability of 2+ Homeruns for Baseball Player in 4 At-Bats

In summary, the formula for calculating the probability of a baseball player hitting 2 or more homeruns out of 4 at bats is 1 - (4 x 0.05 x 0.95^3) - 0.05^4. This can also be seen as a Bernoulli trial problem with a coin toss, where the probability of heads is 0.05. There are 16 possible scenarios for this problem, but there are easier ways to calculate the probabilities than counting them all individually.
  • #1
Muck
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0
The question: A baseball player has a 5% chance to hit a homerun each at bat. If the player is up 4 times, what is the chance he hits 2 or more homeruns. I came up with the answer, but this was a long process. I need a formula. And is there an easier way of counting the possibilities?

I did it like this:

Probability of 0 out of 4 + 1 out of 4 = 1.401875

Probability of 0 out of 4 is (0.95)(0.95)(0.95)(0.95)(1) with 1 possibility

Probability of 1 out of 4 is (0.05)(0.95)(0.95)(0.95)(4) with 4 possibilities so

The other 11 possibly scenarios are 6 to do 2 out of 4, 4 to do 3 out of 4, and 1 to do 1 out of 4, for a total of 16 (4x4) but is there an easier way than counting these?

Thank you!

EDIT: for 'player' :)
 
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  • #2
Your calculations are wrong somewhere. For one, it should be obvious that the first two cases can't have a probability greater than 1, in fact the total probabilities of all the cases should be one. The probability for the first two cases is 0.98598125. Anyways, you can see it was pretty easy to calculate the first two numbers. The probability for 2, 3, or 4 homeruns (i.e. at least 2) is 1 - 0.98598125. You could also calculate it as:

Probability of 2 out of 4 is (0.05)(0.05)(0.95)(0.95)(6) with 6 possibilities

Probability of 3 out of 4 is (0.05)(0.05)(0.05)(0.95)(4) with 4 possibilities

Probability of 4 out of 4 is (0.05)(0.05)(0.05)(0.05)(1) with 1 possibility

Sum those together you'll get the same number as 1 - 0.98598125.
 
  • #3
I wasn't aware that it was the baseball that hit the homerun!


Assuming that any batter has probabilty 0.05 of hitting a homerun at any "at bat", then the probability of a batter hitting k homeruns in n "at bat"s is nCk(0.05)k[/sub](0.95)n-k where nCk is the binomial coefficient: n!/(k!(n-k)!).
 
  • #4
Thank you.
 
  • #5
This is actually a Bernoulli trial problem that can be seen as tossing a coin, where the chance for heads is .05. Since the total probability is 1 = (H+T)^4. We can just look at 1 minus the tail of the series:

1-4HxT^3-T^4.
 

1. How is probability calculated for a baseball player hitting 2+ homeruns in 4 at-bats?

Probability is calculated by dividing the number of desired outcomes (in this case, 2+ homeruns) by the total number of possible outcomes. In baseball, the total number of possible outcomes is determined by the number of at-bats, which is 4 in this scenario. Therefore, the probability is (number of desired outcomes/total number of possible outcomes), or (2/4) = 0.5 or 50%.

2. What factors can affect the probability of a baseball player hitting 2+ homeruns in 4 at-bats?

There are several factors that can affect the probability of a baseball player hitting 2+ homeruns in 4 at-bats. These include the player's skill level, the opposing team's pitcher and defense, the weather and playing conditions, and the player's physical and mental state at the time of the at-bats.

3. Is probability the same for all baseball players?

No, probability is not the same for all baseball players. Each player has their own unique set of skills, strengths, and weaknesses that can affect their probability of hitting 2+ homeruns in 4 at-bats. Additionally, external factors such as the opposing team and playing conditions can also impact probability.

4. Can probability be used to predict future performances of a baseball player?

While probability can be used to estimate the likelihood of a player hitting 2+ homeruns in 4 at-bats, it cannot accurately predict future performances. Probability is based on past data and does not account for variables that may change in the future, such as injuries or changes in the player's skill level.

5. Is probability the only factor that determines a baseball player's success?

No, probability is not the only factor that determines a baseball player's success. While it can provide insight into the likelihood of a player hitting 2+ homeruns in 4 at-bats, there are many other factors that contribute to a player's overall success, including their overall performance, teamwork, and leadership skills.

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