An online calculator to calculate the binomial probability distribution and the probabilities of "at least" and "at most" related to the binomials.
Example 1
At each trial, the probability that event A occurs is \( p = 0.4 \)
a) What is the probability that event A occurs 3 times after 6 trials?
b) What is the probability that event A occurs at least 3 times after 6 trials?
c) What is the probability that event A occurs at most 3 times after 6 trials?
Solution to Example 1
a) \( P(X = 3,6,0.4) = \dfrac{6!}{3! (6-3)!} \cdot 0.4^3 \cdot (1-0.4)^{6-3} = 0.276480 \)
b)
At least 3 times means \( x \) is either \( 3, 4, 5 \; \text{or} \; 6\) or \( x \ge 3 \)
\( P(\text{at least 3 times}) = P( X = 3 \; or \; X = 4 \; or \; X = 5 \; or \; X = 6 ) \)
Using the binomial formula, the probability may be written as
\( P(X \ge 3,6,0.4) = P(X = 3,6,0.4) + P(X = 4,6,0.4) + P(X = 5,6,0.4) + P(X = 6,6,0.4) = 0.455680 \)
c)
At most 3 times means \( x \) is either \( 0, 1, 2 \; \text{or} \; 3\) or \( x \le 3 \)
\( P(\text{at most 3 times}) = P( x = 0 \; or \; x = 1 \; or \; x = 2 \; or \; x = 3 ) \)
Using the binomial formul, the probability may be written as
\( P(X \le 3,6,0.4) = P(X = 0,6,0.4) + P(X = 1,6,0.4) + P(X = 1,6,0.4) + P(X = 3,6,0.4) = 0.820800 \)
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