# Probability distribution for discrete data

• chwala
In summary, the conversation is discussing a textbook problem shared on a WhatsApp group. The problem involves finding the value of k=0.08 and understanding the probability distribution of y. The solution provided by a colleague uses the probability of no chicks surviving, which is a combination of the probability of no eggs, 1 egg, 2 eggs, and so on. The conversation also touches on the use of binomial theorem and conditional probability in solving similar problems with different probability distributions. The general pattern of the problem involves partitioning the probability space and applying the concept of conditional probability.
chwala
Gold Member
Homework Statement
For the attached problem below, write the probability distribution of (ii.) the number of chicks of chicks surviving per pair of adults.
Relevant Equations
probability distribution for discrete data

this is a textbook problem shared on a whattsap group by a colleague...
i have no problem in finding the value of ##k=0.08##, i have a problem with part (ii) of the problem. I have attached the solution here;

how did they arrive at the probability distribution of ##y##?

attached below is working from a colleague, i do not understand how he arrived at the solution. What rule is being used here?

The probability that no chicks survive is the probability of no eggs (0.2), plus the probability of 1 egg times the probability that it does not survive (0.24 * (1-0.8)), plus the probability of two eggs times the probability that neither survives (0.32 * (1-0.6)2), etc. You get the pattern?

chwala
mjc123 said:
The probability that no chicks survive is the probability of no eggs (0.2), plus the probability of 1 egg times the probability that it does not survive (0.24 * (1-0.8)), plus the probability of two eggs times the probability that neither survives (0.32 * (1-0.6)2), etc. You get the pattern?

yes, very interesting indeed...so there is no specific rule to this? i thought i could use binomial theorem but i guess i was wrong...what about the pattern in ##p(1)##, a bit confusing there on ##0.32*2*0.6*0.4##

This question is a bit confusing to me, we have the probability distribution of the laying of the eggs ...i.e
##[0.2, 0.24, 0.32, 0.24, 0]## for ##[x=0, 1,2,3,4]## , now we are given the probabilities of the survival ...ie ##[0.8, 0.6,0.4]## for ##[x=1,2,3]##, my question is , is there a relationship in coming up with probability distribution for similar problems where the probability distribution , call it ##x## and probability values of another variable, call it ##y## are given...so as to find the probability distribution of ##y##, call this distribution ##Y##
is this not conditional probability in a way?
very interesting question, i think i now get it, one has to think in terms of combination in order to attempt this question step by step...cheers

Last edited:
chwala said:
what about the pattern in p(1), a bit confusing there on 0.32*2*0.6*0.4
That's where your binomial theorem comes in. Probability of k successes out of n is
nCkPk(1-P)n-k
So P(2 eggs, 1 survives) = P(2 eggs)*2*P(survive)*P(don't survive)
Factor 2 because either egg might be the one that survives.

true, i can see now...cheers

My colleagues suggest that we cannot apply binomial distribution the way that I have done, for the simple reason that there are conditions to be followed...##p## and ##q## has to be a fixed or rather a constant value throughout...when using binomial distribution...

You can't use the same binomial distribution for the whole thing because the survival probabilities are not the same. But for one specific contributor to the probability (e.g. the probability of 2 eggs out of 3 surviving) you use the binomial distribution with the survival probability value for that situation.

chwala
( The ornithologist is apparently assuming the survival of each chick in an nest with k eggs is independent of the survival of the other chicks. )

As to the general pattern of the problem. Suppose a probability space is partitioned in M+1 mutually exclusive events ##C_0,C_`1,...C_{M+1}##

The probability of any given event S in the same probability space satistifes:

##Pr(S)=Pr(S∩C_0)+Pr(S∩C_1)+...Pr(S∩C_{M+1})##

## = Pr(S | C_0) Pr(C_0) + Pr(S | C_1) Pr(C_1) + ...Pr(S |C_{M+1}) Pr(C_{M+1})##In the problem, we can let ##C_k=## "k eggs are laid".

For a particular event such as S= "2 chicks survive", you can apply the above expression. A factor such as ##P(S|C_3)## is given by the probability of 2 successes of a random variable that has a binomial distribution with 3 trials and probability of success 0.4.

chwala said:
my question is , is there a relationship in coming up with probability distribution for similar problems where the probability distribution , call it x and probability values of another variable, call it y are given...so as to find the probability distribution of y, call this distribution Y
is this not conditional probability in a way?

As you can see from the above discussion, conditional probability is part of the general pattern. The appearance of binomial distributions is special feature of this particular problem.

chwala

## What is a probability distribution for discrete data?

A probability distribution for discrete data is a statistical function that shows the likelihood of different outcomes occurring in a discrete data set. It assigns a probability to each possible value or range of values in the data set.

## How is a probability distribution for discrete data different from a continuous data distribution?

A probability distribution for discrete data is different from a continuous data distribution in that it only applies to data that can take on a finite or countable number of values. Continuous data, on the other hand, can take on an infinite number of values within a given range.

## What are some examples of discrete data?

Examples of discrete data include the number of siblings a person has, the number of cars in a parking lot, the number of heads in a coin toss, and the number of students in a classroom.

## How is a probability distribution for discrete data represented?

A probability distribution for discrete data is typically represented using a probability mass function (PMF) or a cumulative distribution function (CDF). The PMF shows the probability of each individual value occurring, while the CDF shows the probability of a value being less than or equal to a certain threshold.

## What is the importance of understanding probability distribution for discrete data?

Understanding probability distribution for discrete data is important in many fields, including statistics, economics, and engineering. It allows us to make predictions and decisions based on the likelihood of different outcomes occurring in a given data set. It also helps us to understand and analyze data, as well as to make informed decisions and draw conclusions from the data.

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