SOLUTION: 55% of all shark attacks are not fatal. Let r be the number of nonfatal attacks out of a random sample of five shark attacks. a) What is the probability all five shark attacks are

Click here to see ALL problems on Probability-and-statistics

Question 1209960: 55% of all shark attacks are not fatal. Let r be the number of nonfatal attacks out of a random sample of five shark attacks. a) What is the probability all five shark attacks are nonfatal? b) What is the probability that three or more of the five shark attacks are nonfatal? c) What is the expected number of nonfatal shark attacks out of the five? d) What is the standard deviation of the r-probability distribution?

Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959) (Show Source):
You can put this solution on YOUR website!
Let's solve this problem step-by-step.
**Understanding the Problem**
This is a binomial probability problem because:
* There are a fixed number of trials (n = 5).
* Each trial has only two possible outcomes (nonfatal or fatal).
* The probability of success (nonfatal) is constant (p = 0.55).
* The trials are independent.
**Given Information**
* Probability of nonfatal shark attack (success), p = 0.55
* Probability of fatal shark attack (failure), q = 1 - p = 1 - 0.55 = 0.45
* Number of trials (sample size), n = 5
**Formula for Binomial Probability**
The probability of getting exactly k successes in n trials is:
P(X = k) = (nCk) * p^k * q^(n-k)
where:
* nCk = n! / (k! * (n-k)!) (the number of combinations of n items taken k at a time)
**a) Probability All Five Shark Attacks Are Nonfatal (k = 5)**
P(X = 5) = (5C5) * (0.55)^5 * (0.45)^(5-5)
P(X = 5) = 1 * (0.55)^5 * (0.45)^0
P(X = 5) = (0.55)^5 * 1
P(X = 5) ≈ 0.0503284375
**b) Probability Three or More of the Five Shark Attacks Are Nonfatal (k ≥ 3)**
We need to calculate P(X = 3), P(X = 4), and P(X = 5), then add them up.
* P(X = 3) = (5C3) * (0.55)^3 * (0.45)^(5-3) = 10 * (0.55)^3 * (0.45)^2 ≈ 0.275653125
* P(X = 4) = (5C4) * (0.55)^4 * (0.45)^(5-4) = 5 * (0.55)^4 * (0.45)^1 ≈ 0.2058890625
* P(X = 5) ≈ 0.0503284375 (calculated in part a)
P(X ≥ 3) = P(X = 3) + P(X = 4) + P(X = 5)
P(X ≥ 3) ≈ 0.275653125 + 0.2058890625 + 0.0503284375
P(X ≥ 3) ≈ 0.531870625
**c) Expected Number of Nonfatal Shark Attacks (Mean)**
For a binomial distribution, the expected number (mean) is:
* μ = n * p
* μ = 5 * 0.55
* μ = 2.75
**d) Standard Deviation of the r-Probability Distribution**
For a binomial distribution, the standard deviation is:
* σ = √(n * p * q)
* σ = √(5 * 0.55 * 0.45)
* σ = √(1.2375)
* σ ≈ 1.11242528
**Answers:**
a) Approximately 0.0503
b) Approximately 0.5319
c) 2.75
d) Approximately 1.1124

Answer by ikleyn(52782) (Show Source):
You can put this solution on YOUR website!
.
55% of all shark attacks are not fatal. Let r be the number of nonfatal attacks out of a random sample of five shark attacks.
(a) What is the probability all five shark attacks are nonfatal?
(b) What is the probability that three or more of the five shark attacks are nonfatal?
(c) What is the expected number of nonfatal shark attacks out of the five?
(d) What is the standard deviation of the r-probability distribution?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Be aware !

Calculations and the answer in the post by @CPhill in part (b) are INCORRECT.