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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
Etiology is the study of the cause of a disease.
The probability this student will fail is the same as the probability that some other student will flip a fair coin 20 times and get less than 8 heads, i.e., more than 12 tails.
There are 2^20 possible different-looking sequences of 20 coinflips, which we assume all have equal probability.
Of those sequences, 1 has no heads at all, 20 have exactly 1 head, 190 have exactly 2 heads, ... and 77520 have exactly 7 heads.
So we sum up all those possible ways to fail and we get ...
... ...
I'm assuming that the student answers randomly, flipping a fair (50:50) coin on each question to choose "true" or "false".
In that special case, it doesn't matter how many of those twenty questions are true or how many are false.
(If the student answers randomly by flipping an unfair coin, say a 25:75 coin, then it does matter how many of those questions are true -- I'll let you figure that one out).
false.
Empirical probabilities.
What is the probability of rolling a number higher than 4?
There are six faces, and only three faces with a number higher than a four. So it is 3 chances in 6.
What are the probability of rolling a prime number on one roll of a die?
1 out of 2 if the die is six-sided.
What are the 6 best numbers to use to win any lottery?
There is no set of "best numbers". If there were someone would have used them again and again to amass a fortune.
When tossing 5 coins simultaneously find the probability that at least 1 head is showing.?
Assuming unbiased coins which won't land and stay on their edge, the probability of head = probability of tail = ½
→ probability no heads = probability 5 tails = ½^5 = 1/32
→ probability of at least one head = 1 - 1/32 = 31/32 = 0.96875 = 96.875 % = 96 7/8 %
Why is only one normal distribution table needed to find any probability under the normal curve?
As the distribution is symmetric about its mean, only tables up to the mean need be calculated/given in a table. The area under the curve between any two points can then be calculated.
For a normal distribution with mean µ and standard deviation σ, a z value is calculated for a given point x:
z = (x - µ) / σ
This z value is then used to look up the area in the given "half" tables, giving the area (probability) of the value lying between the mean and the given z value. If negative, z is below the mean, but for the table, the sign is ignored.
This can be expressed as:
area = normal(|z|)
where normal(z) is the value in the normal table at the given (positive) z value.
To calculate the area between two points (ie the probability that a value lies between two given values), their corresponding z values (z₁ and z₂) are first calculated and then combined viz:
- If they are both on the same side of the mean (ie z₁ and z₂ have the same sign) then the area is given by:
- If they are on opposite sides of the mean (ie z₁ and z₂ have different signs) then the area is given by:
Almost all of the normal distribution lies between ±4 standard deviations of the mean.
What is the difference between experimental and theoretical physics?
When theoretical physicists work on equations and don't test their hypothesis, experimental physicists test their hypothesis and verify their conclusion. Usually theoretical physicists work on things like black holes and string-theory when experimental physicists work on Newtonian laws.
What is the probability of 3 students out of 11 getting the same grade on an exam?
It is not possible to answer the question since there is no information on how many grades there are. Second, the probability of a student getting a particular grade will depend on how "wide" the grade is: some grade boundaries are closer together than others. Furthermore, the grades achieved by the students are not independent events. If the exam is very easy, they might all get a top grade.
How are ratio and probability the same?
A ratio is a comparison of the relative size of two different things. Probability is the change that something will (or will not) occur.
Probability can be expressed as a ratio of Yes to No (or, "will occur" to "won't occur"). That is, Probability is the relative size of Yes to No. So, if something is said to have a 60% Probability of occurring, what that is indicating is that, out of 100 tries, 60 will be the outcome indicated. While probability is usually expressed as a percentage, it is entirely possible to express it as a ratio. In the aforementioned example, a 60% Probability of occurrence could also be said to be a 60:40 (or, reduced, 3:2) ratio in favor of happening.
The probability that all balls are red is 0.50 or 50%.
EXPLANATION
Suppose there are 4 boxes with 5 balls each.
Box A has 2 red balls.
Box B has 3 red balls.
Box C has 4 red balls.
Box D has 5 red balls.
The probability of drawing at random 2 red balls for each box is:
Box A; P(2 red balls) = (2/5)∙(1/4) = 2/20 = 1/10
Box B; P(2 red balls) = (3/5)∙(2/4) = 6/20 = 3/10
Box C; P(2 red balls) = (4/5)∙(3/4) = 12/20 = 6/10
Box D; P(2 red balls) = (5/5)∙(4/4) = 20/20 = 10/10
Now, suppose we do the drawing of 2 balls experiment 10 times on each box
giving a total of 40 experiments. The probabilities calculated above are the
"expected" results, that is; out of the 40 experiments (drawing of 2 balls), two
red balls resulted:
1 time came from box A
3 times from box B
6 times from box C
10 times from box D
Notice that from the 40 experiments, 20 result in 2 red balls.
From here we have that when drawing 2 red balls the probability that it came
from a box containing 2 red balls (box A), 3 red balls (box B), 4 red balls (box C)
or 5 red balls is:
P(A) = 1/20 = 0.05 = 5%
P(B) = 3/20 = 0.15 = 15%
P(C) = 6/20 = 0.30 = 30%
P(D) = 10/20 = 0.50 = 50%
What is an example of independence events?
An example of independent events is rolling a die and flipping a coin. The roll of the die has no affect on the flipping of the coin and the converse is true also. Another example is the rolling of a die followed by the rolling of a die - they are sequentially unrelated - so they are independent.
The outcomes for each set is not independent. For a set to go into a tie breaker the players need to be very evenly matched. If they are, there is a greater probability that the sets will go into tie breaker. If you disregard that aspect of realitythen the answer is 0.0270, approx.
A group's reported are more likely to become equal to the statistical probability of loss?
the larger the group, the more likely the statistical probability of loss will be equal
