A biologist estimates that 80% of the deer in a region carry a certain type of tick. For a sample of 300 deer selected at random what is the chance that 246 or fewer deer have this tick?
To solve this problem, we will use the binomial probability formula. The formula for the binomial probability is:
P(x) = C(n, x) * p^x * (1-p)^(n-x)
Where:
P(x) represents the probability of getting exactly x successes,
n represents the total number of trials or observations,
x represents the number of desired successes,
p represents the probability of success in a single trial.
In this scenario:
n = 300 (number of deer in the sample)
p = 0.8 (the probability of a deer carrying the tick)
x = 246 or fewer (the desired number of deer)
Now, we need to calculate the probability of getting 246 or fewer deer carrying the tick. We will sum up the individual probabilities for each possible value of x.
P(246 or fewer) = P(0) + P(1) + P(2) +...+ P(246)
To calculate this probability, we'll use a calculator, spreadsheet, or a statistical software. Alternatively, we can use a cumulative binomial distribution table.
Let's calculate the probability using a calculator.
To calculate the probability that 246 or fewer deer have the tick, given that 80% of the deer in the region carry the tick and a sample of 300 deer is selected at random, you can use the binomial distribution formula.
The binomial distribution is used when you have a fixed number of independent trials with only two possible outcomes (in this case, either a deer carries the tick or it does not). The formula is as follows:
P(X ≤ k) = Σ (nCk * p^k * q^(n-k))
Where:
P(X ≤ k) is the probability of getting k or fewer successes
n is the total number of trials (in this case, 300)
k is the number of successful outcomes (deers with the tick)
p is the probability of a successful outcome (80% or 0.8, in decimal form)
q is the probability of failure (100% - p, or 20% or 0.2, in decimal form)
nCk is the number of possible combinations of k successes from n trials (n choose k).
To calculate the probability for 246 or fewer deer with the tick, you need to sum up the probabilities for each possible number of successes (from 0 to 246).
Calculating the probability for each value of k individually would be time-consuming, so I will use a statistical software or calculator to perform the calculations.