Final answer:
To calculate P(x≤6) for a binomial distribution with n=9 and p=0.8, one can use the binomial cumulative distribution function or a calculator's statistical functions to sum the probabilities of x successes for x being 6 or fewer.
Step-by-step explanation:
The question pertains to calculating the probability of a binomial random variable being less than or equal to a certain value. Given n=9 trials and a success probability of p=0.8 in a binomial distribution, we are interested in finding P(x≤6), or the cumulative probability of x successes where x is 6 or fewer. This calculation requires using either a binomial probability formula, a statistical table, or technology like a calculator with binomial distribution functions.
To compute P(x≤6) for the binomial variable X with n=9 and p=0.8, one would typically use the binomial cumulative distribution function (cdf). The formula for the binomial probability of exactly x successes in n trials is P(X = x) = (n choose x) * px * (1-p)n-x. However, to find the cumulative probability P(x≤6), you would sum the probabilities from P(X=0) to P(X=6), or use a function like binomcdf on a calculator if available.
The binomial distribution helps us understand the likelihood of a specific number of successes (x) in a series of independent trials (n), with a constant success probability (p) for each trial. In our case, P(x) for a binomial random variable with parameters n and p can be efficiently computed using modern calculators with statistical functions or software that includes the binomial distribution calculations.