Solve Geometric Probability Problems
The probabilities involved in geometric problem that also called as geometric probability
. Geometry probability may be circle or any polygon from the geometric. It involves the length, area and volume of any one of geometric shapes.
The definition of probability is
Number of successful outcomes
Probability = _______________________________
Total number of possible outcomes.
Two types of geometric probability:
Theoretical probability
Empirical probability
Example for Geometric Probability Problems:
Solve the following geometric probability problems.
pro 1: Distribute the picture papers clips and demonstrate how to rotate the picture paper clip to the corners. Moreover, each pair of students rotates to another person that change to 48 times and noted their results in a table. Collect the data from the field on the number of times the changed one another place and that has stop in each of the fields. Find the probability of rotation between P, Q, R, or S from the data.
Sol 1: By geometric probability formula we can solve the above problem.
The picture papers are rotate different way that is P, Q, R, or S.
The probability for given data has identified has four field.
The circle has portion of 180, 90, 60, 30 degrees, total number of degrees in the circle, area of the sector of the circle has given.
The probability of given problem is = 48/ 4
=12
Check this
Bivariate Normal Distribution Example awesome i recently used to see.
Pro 2: Distribute the ball to everyone in the pentagon and demonstrate how to change one to another pentagon sides and that total has a five corners. Moreover, each of corners has change the ball one to another person and that ball has rotated 45 times and noted their results in a table. Collect the data from the field on the number of times the ball lands in each of the fields. Find the probability of spinning A, B, C, D, and E from the data.
Sol 2: By geometric probability formula we can solve the above problem.
Here the pentagon has the five corners such as A, B, C, D, E
The pentagon will be rotated into 45 times,
We have to find the probability here,
Probability = Number of successful outcomes / total number of possible outcomes probability =45/5 =9
So the probability of given problem is 9
Probability of success is a matter of concern from ancient times whether success is just confined to getting a Heads in tossing a coin or getting a triplet of aces (including that of spade) from randomly selecting just three cards from well shuffled deck of 52 cards! or even more to estimate the overall power of enemy army.
Success Vs Probability
Success is defined as the required or favorable outcome or combination of outcomes (known as event) out of many possible outcomes of the Random Experiment. There can be atleast two cases of this situation. One is all the possible outcomes of the Random Experiment are equally likely to occur
or
there is a criteria on which the probability is defined for each possible outcome separately. Based on these cases or facts the probability of success is calculated. To make this point more clear, lets take an example of most popular random experiment of tossing a coin. This experiment has two possible outcomes i.e. either the coin lands with heads up or with tails up. Now the case first is that the possible outcomes are equally likely. That means if we consider the success as the coin landing with heads up, then the success as well as the failure (the coin landing with tails up) are both having the same number of chances (probability) to occur. Since there are only two elementary events, thus the probability of success is 1/2 or 50%.
Now the second case is when the outcomes are not equally likely. In that case we assume that the probability of success be p (here success is event of the coin landing with heads up) and consequently due to the fact that one of the possible outcomes has to occur (mutually exhaustive events) and both the outcomes cannot occur simultaneously (mutually exclusive events) and also that the probability of sure event is 1 (sample space: here Heads up or Tails up), we have the probability of failure as 1-p.
Probability of Multiple Successes
Its just well known that calculating the probability for just single success is not enough. Since the matter of concern in our daily life is to calculate the minimum number of trials of a given Random Experiment to get a given number of successes. Or the problem can be to find the probability of getting a predeclared number of successes in a given number of trials. For that we have a special formula called Binomial Distribution. In this, success and failure are not taken equally likely events as this case can be easily derived as the special case. B(n,p) is used as notation of binomial distribution with n number of trials and probability of success being p.
According to this law, the probability of getting r number of successes can be calculated by finding the coefficient of xr in the expansion of (q+px)n where q=1-p is the probability of failure. Thus, P(x)=C(n,r)prqn-r where C(n,r) is combinatorial term of selecting r items out of n items. Studying probability of success is hot and happening subject throughout the world. The lottery system, casino, traffic jam, are some application of interest to name a few. Here I do want to mention a probability problem called Monty Hall Paradox based on US Television Show "Lets Make A Deal" !!
by: mathqa
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