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Fido ate 52 containers of ice cream in 13 different flavors, with 4 of the container containing each flavor. If the first 5 containers were chosen at random, without replacement, what are the chances that 4 of those containers are the same flavor.

Sagot :

Using the hypergeometric distribution, it is found that there is a 0.0002 = 0.02% probability that 4 of those containers are the same flavor.

The flavors are chosen without replacement, which means that the hypergeometric distribution is used to solve this question.

Hypergeometric distribution:

[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

The parameters are:

  • x is the number of successes.
  • N is the size of the population.
  • n is the size of the sample.
  • k is the total number of desired outcomes.

In this problem:

  • There are 52 containers, hence [tex]N = 52[/tex]
  • For each flavor, there are 4 containers, hence [tex]k = 4[/tex]
  • 5 containers are chosen, hence [tex]n = 5[/tex]

The probability of choosing all containers of one flavors is P(X = 4).

  • There are 13 containers, hence, the probability is multiplied by 13.

[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]

[tex]P(X = 4) = h(4,52,5,4) = \frac{C_{4,4}C_{48,1}}{C_{52,5}} = 0.000018469[/tex]

13 x 0.000018469 = 0.0002

0.0002 = 0.02% probability that 4 of those containers are the same flavor.

A similar problem is given at https://brainly.com/question/25743676

The probability that 4 of those containers are the same flavor is 0.02% and this can be determined by using the Hypergeometric distribution.

Given :

  • Fido ate 52 containers of ice cream in 13 different flavors, with 4 containers containing each flavor.
  • The first 5 containers were chosen at random, without replacement.

The probability according to the Hypergeometric distribution is given by:

[tex]\rm P(X=x) = h(x,N,n,k)=\dfrac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]     --- (1)

where the total number of successes is 'x', population size is 'N', the sample size is 'n', and the total number of desired outcomes is 'k'.

Now, substitute the values of known terms in the expression (1).

[tex]\rm P(X=4) = h(4,52,5,4)=\dfrac{C_{4,4}C_{48,1}}{C_{52,5}}[/tex]

[tex]\rm P(X=4) =0.000018469[/tex]

Now, multiply the above result by 13.

[tex]13\times 0.000018469=0.0002[/tex]

So, the probability that 4 of those containers are the same flavor is 0.02%.

For more information, refer to the link given below:

https://brainly.com/question/23017717

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