Question

if the probability that a tourist will be bitten by a rattlesnake is 0.00002, what is the probability that among 75,000 tourists visiting the grand canyon at least 2 will be bitten by a rattlesnake?

Answer 1

To solve this problem, we can use the binomial distribution, which is a probability distribution that describes the number of successes (in this case, being bitten by a rattlesnake) in a fixed number of independent trials (in this case, the number of tourists visiting the Grand Canyon). The formula for the probability mass function of the binomial distribution is:

P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)

where:

X is the number of successes

k is the number of successes we're interested in (in this case, at least 2)

n is the total number of trials (in this case, 75,000)

p is the probability of success in each trial (in this case, 0.00002)

(n choose k) is the binomial coefficient, which is equal to n! / (k! * (n - k)!)

To find the probability of at least 2 tourists being bitten by a rattlesnake, we need to calculate the probabilities of exactly 2, exactly 3, and so on, up to 75,000. However, this would be very time-consuming and impractical. Instead, we can use the complement rule, which says that the probability of an event occurring is equal to 1 minus the probability of the event not occurring. In this case, the event we're interested in is at least 2 tourists being bitten by a rattlesnake, so the complement of this event is no tourists or only 1 tourist being bitten.

The probability of no tourists being bitten is:

P(X = 0) = (75,000 choose 0) * 0.00002^0 * (1 - 0.00002)^(75,000 - 0)

= 1 * 1 * 0.99998^75,000

≈ 0.99928

The probability of only 1 tourist being bitten is:

P(X = 1) = (75,000 choose 1) * 0.00002^1 * (1 - 0.00002)^(75,000 - 1)

≈ 0.000717

So, the probability of at least 2 tourists being bitten is:

P(X >= 2) = 1 - P(X = 0) - P(X = 1)

≈ 1 - 0.99928 - 0.000717

≈ 0.000003

Therefore, the probability that among 75,000 tourists visiting the Grand Canyon at least 2 will be bitten by a rattlesnake is approximately 0.000003, or about 0.0003%.