Solving the Trinomial & Binomial Distributions: A Challenge

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The discussion focuses on deriving a formula for the Trinomial distribution, which accounts for three possible outcomes in an experiment: success, fail, and unknown, with respective probabilities p, q, and r. Participants are asked to solve specific probability problems related to an indicator light system with three states: OFF, FLASHING, and ON. The probabilities for these states are given as 1/2, 2/5, and 1/10, respectively, and the challenge includes calculating the probabilities of various combinations of light states across five independent lights. The request emphasizes the need for solutions to be expressed as fractions and encourages participants to show their work according to forum guidelines. The thread highlights the complexities of both the Trinomial distribution and practical applications in probability scenarios.
engr.humayun
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can anyone help me please

can anyone solve this problem for me please

Q) The Binomial distribution allows the calculation of the probability of k successes in n trails where there are only two outcomes: success or fail with probabilities p and q respectively. The Binomial probability is given by

n! / (n-k)! * p k q n-k ( / is division sign)

a) Consider a case where an experiment has 3 possible outcomes (success, fail, unknown) with probabilities p , q and r respectively. Derive a formula for the Trinomial distribution i.e. the probability of getting k success, j fail and (n-(k+j)) unknown outcomes from a sequence of n trails.

b) An indicator light can be in one of three states: OFF, FLASHING and ON, with probabilities 1/ 2 , 2 /5 and 1 /10 respectively. A test panel has 5 such lights whose states are mutually independent.

i. What is the probability that all five lights are OFF?
ii. What is the probability that three lights are OFF, one light is FLASHING and one light
is ON?
iii. What is the probability that three or more lights are OFF and at most one is ON?

Express all results as fractions.
 
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Can you please show your attempt at a solution as per the PF guidelines.

Jared
 

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