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Probability

The probability of a certain event is a number expressing the likelihood that a specific event will occur, expressed as the ratio of the number of actual occurrences to the number of possible occurrences. In mathematics, it is a measure of how often an event will happen and is the basis of statistics.

14,643 Questions

How can you prove the combination formula?

Combination Formula Proof

GENERIC:

  • Let C(n,r) be the number of ways to generate unordered combinations
  • The number of ordered combinations (i.e. r-permutations) is P(n,r)
  • The number of ways to order a single one of those r-permutations P(r,r)
  • The total number of unordered combinations is the total number of ordered combinations (i.e. r-permutations) divided by the number of ways to order each combination
  • Thus, C(n,r) = P(n,r)/P(r,r) = [n!/(n-r)!]/r!/(r-r)!] = n!/r!(n(n-r)!

SPECIFIC:

  • Let C(52,5) be the number of ways to generate unordered poker hands
  • The number of ordered poker hands is P(52,5) = 311,875,200
  • The number of ways to order a single poker hand is P(5,5) = 5! = 120
  • The total number of unordered poker hands is the total number of ordered hands divided by the number of ways to order each hand
  • Thus, C(52,5) = P(52,5)/P(5,5)

Discrete and Continuous data is used in mathematics?

Yes it is. Discrete data is something that's set. Like say you were making a line graph about renting bikes. You can only rent whole bikes there is nothing in between. You shouldn't connect points on a line graph with discrete data but some cases can be argued.

Continuous data is usually a measurement that could change like time.

What is the probability that a family of four children has at least 2 girls?

Going by the assumption that the probability of a child being a girl is 1/2, and that the events are independent, then the probabilities of different numbers of girls are as follows:

P(0 girls) = 1/2 ^ 4 = 1/16

P(1 girl) = 4 * (1/2) * (1/2 ^ 3) = 1/4

P(2 girls) = 6 * (1/2 ^ 2) * (1/2 ^ 2) = 3/8

P(3 girls) = 4 * (1/2 ^ 3) * (1/2) = 1/4

P(4 girls) = 1/2 ^ 4 = 1/16

Therefore the probability of at least 2 girls = 3/8 + 1/4 + 1/16 = 11/16

Two coins are tossed What is the probability of both coins are heads?

The probability that both coins are heads is the probability of one coin landing heads multiplied by the probability of the second coin landing heads: (.5) * (.5) = .25 or (1/2) * (1/2) = 1/4

I am given 9 15 28 10 and 8 what is the mean absolute deviation for this data?

The mean deviation or absolute mean deviation is the sum of the differences between data values and the mean, divided by the count. In this case the MAD is 6.

What does mutally exclusive mean?

It means the two events cannot simultaneously occur; for example the two events, being dead and being alive are mutually exclusive, since they cannot occur at the same time.

The odds in favor of an event are 10 to1 ratio Find the probability that the event will occur?

If the odd favoring an event are 10 to 1, then the probability of the event occurring is 0.9.

The odds in favor of an event are 10:1.

Find the probability that the event will occur.

----

P(E)+P(E') = 1

---

P(E)/P(E') = 10/1

So P(E) = 10P(E')

----

Substitute for P(E) and solve for P(E'):

10P(E')+P(E') = 1

11P(E') = 1

P(E') = 1/11

---

Therefore P(E) = 10/11

How many marbles can you fit in a jar?

That depends on the size of the marbles, as well as the size and shape of the jar.

The Central Limit Theorem is important in statistics because?

According to the central limit theorem, as the sample size gets larger, the sampling distribution becomes closer to the Gaussian (Normal) regardless of the distribution of the original population. Equivalently, the sampling distribution of the means of a number of samples also becomes closer to the Gaussian distribution. This is the justification for using the Gaussian distribution for statistical procedures such as estimation and hypothesis testing.

For a normal distribution find the z-scores values that separate the middle 60 percent of the distribution for the 40 percent in the tails?

The standard normal table tells us the area under a normal curve to the left of a number z.

The tables usually give only the positive value since one can use symmetry to find the corresponding negative values.

The middle 60 percent leaves 20 percent on either side. So we want the z scores that correspond to that 80 percentile which is .804. Therefore the values are are between z scores of -.804 and .804

* * * * *

I make it -0.8416 to 0.8416

What is the element of the sample space for a coin then rolling a die?

It is an ordered pair of the form (A, n) where A is the outcome of the tossed coin (H or T) and n is the outcome of the rolled die (1, 2, 3, 4, 5, 6).

Three fair dice are tossed What is the probability that at least two of them show that same face?

Total number of outcomes = 6 x 6 x 6 = 216

Number of ways that are successful:

1 1 -

1 - 1

- 1 1

1 1 1

2 2 -

2 - 2

- 2 2

2 2 2

3 3 -

3 - 3

- 3 3

3 3 3

etc.

Four ways for each number 1 thru 6 = 24 possible successes. But for each of the one's with a dash (-), there are 5 outcomes (121, 131, 141, 151, 161, 211, 311,411,...etc), so the real number of ways for this to happen is 16 ways for each number: 6 x 16 = 96.

Probability of success = 96/216 = 4/9 = 44.4% (rounded)

What are the chances of dealing out cards that have been shuffled well in exact order?

The odds are... remote. This is a factorial problem to calculate the answer you take how many options the first "dealt" card has of being. In this case the first card has a 1-in-52 chance of being the Ace of Spades (the "highest" card in a deck.) The next card has a 1-in-51 chance of being the Two of Spades, and so on. You calculate the odds of this happening my multiplying 52 by 51.

So, the odds are 52 times 51 times 50 times 49 times 48 times... yadda yadda yadda until you get to one. The answer?

The odd of taking a shuffled deck of cards, reshuffling it, and the dealing the cards back in the correct order are 1 in 806,581,751,709,439x10 to the 54th power. (septendecillion.)

Or 80 unvigintillion. (80 followed by 66 zeroes). For the sake of comparison, conservative estimates put the number of stars in our galaxy at 100 Billion (100 followed by 9 zeroes) and the number of galaxies in the universe at the same number, making the number of potential stars in the universe at 10 sextilion (a 10 followed by 21 zeroes.)

A bag contains 3 red marbles 2 blue marbles and 5 green marbles What is the probability of selecting a red marble without replacing it in the bag and then selecting a green marble?

Total marbles in the bag = 10

Number of red ones = 3

Probability of pulling a red one on the first draw = 3/10 = 0.3

Total marbles remaining after the first draw = 9

Number of green ones = 5

Probability of pulling a green one after a red one has been withdrawn = 5/9

Probability of both outcomes = (3/10) x (5/9) = (15/90) = 1/6 = (16 and 2/3) percent.

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