Probability

Probability and Statistics (Part 2)

By Madison Nef

How to calculate probability

In the case of a die, there are six equally likely outcomes that can occur.  A collection of outcomes is an event. In the case of a die, the chance of rolling an even number {2,4,6} is classified as an event, since there is more than a 1/6 chance of rolling a certain number. Therefore:

The probability of rolling a die and getting any number in general is 1.
The probability of rolling a die and getting nothing is 0.
The probability of rolling a die and getting a 5 is 1/6, so the probability of NOT getting a 5 is 5/6.
The probability of getting a 2, 4, or 6 in a roll is 3/6, simplified to ½.

Probability in a nutshell is a fraction or a percentage that is calculated in a very simple formula: Probability(e) is the simplified fraction and/or percentage created when you use the total number of possible outcomes as the denominator and the number of outcomes in e as the numerator. So in the case of a die, as we know, the TOTAL possible number of outcomes is 6. So 6 would be the denominator. Let’s say that we are trying to calculate the probability of getting a 2, 5, 3, and 1. There are FOUR outcomes that we are trying to figure, so by using four as the numerator and six as the denominator, we can see that the chances of getting any of those numbers {2,5,3,1} is 4/6, or 2/3.

I know I said this in my last report, but cards are excellent examples of probability. An example used in the lecture was taking a deck of cards and calculating the probability of getting a certain hand of 5 cards in a poker game. The hand that was tested was a hand where all 4 aces are present along with one other card. Calculating how many hands you can have totally from one deck of cards does not give an accurate answer… because you only need one extra card. However, by switching the order of just one card, you create a completely different hand… so the initial answer given is in-accurate. The actual stats for how many different 5-card hands you can get from a deck of 52 cards are about 1,120,000 different hands.

How to figure out the statistics in a deck of cards

1)      What is the probability of being dealt a hand with 4 aces?

2)      Since there are 52 cards in a deck and 4 of those need to be in the hand, we subtract 4 from 52 to get a total of 48 possible cards that could count as the fifth card.

3)      We then figure out how many possible 5-card hands there are: 52x51x50x49x48=311,875,200

4)      However, this number is incorrect since the cards will have repeats… just in a different order. This, in poker, makes and entirely different hand… so we calculate how many different times a hand could be repeated in a different order: 5x4x3x2x1=120

5)      From this, we know that the distinct number of 5 card hands we could get must be the number of possible 5 card hands divided by the number of times a hand could be repeated… 311,875,200/120 = 2,598,960

6)      Finally, we divide 48 by this number to get a final answer of 0.00002% chance of getting four aces in one hand.