Question

The shell of the land snail Limocolaria martensiana has two possible color forms: streaked and pallid. In a certain population of these snails, \(60 \%\) of the individuals have streaked shells. \({ }^{19}\) Suppose that a random sample of 10 snails is to be chosen from this population. Find the probability that the percentage of streaked-shelled snails in the sample will be (a) \(50 \%\) (b) \(60 \%\). (c) \(70 \%\)

Short answer

The probability of getting a 50% streaked sample is \( P(5/10) \), 60% is \( P(6/10) \), and 70% is \( P(7/10) \), where P(k/10) is calculated using the binomial formula.

Step-by-step solution

- Understand the Distribution

The problem suggests that we can treat the color form distribution of the snail shells as a binomial distribution, where the probability (p) of finding a snail with a streaked shell is 0.60. In a binomial distribution, to find the probability of exactly k successes (streaked shells) in n trials (snails), we use the formula: \( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( \binom{n}{k} \) is the binomial coefficient.

- Calculate the Probability for (a) 50%

For (a), we want the probability that exactly 5 out of 10 snails have streaked shells. Using the binomial formula: \( P(X=5) = \binom{10}{5} (0.60)^5 (0.40)^5 \). We compute the binomial coefficient \( \binom{10}{5} = 252 \) and then calculate the probability.

- Calculate the Probability for (b) 60%

For (b), we want the probability that exactly 6 out of 10 snails have streaked shells. So we use the binomial formula with k=6: \( P(X=6) = \binom{10}{6} (0.60)^6 (0.40)^4 \). We compute the binomial coefficient \( \binom{10}{6} \) and calculate the probability.

- Calculate the Probability for (c) 70%

For (c), we are looking for the probability that exactly 7 out of 10 snails have streaked shells. We apply the binomial formula with k=7: \( P(X=7) = \binom{10}{7} (0.60)^7 (0.40)^3 \). We calculate the binomial coefficient \( \binom{10}{7} \) and the probability.

- Compute the Mathematical Calculations

Perform the actual calculations for each case. To facilitate, you can use a calculator or computational software to find the binomial coefficients and compute the powers of p and (1-p).

Key concepts

Binomial Coefficient

To understand the binomial distribution probability, one must first grasp the concept of a binomial coefficient. A binomial coefficient, denoted as \( \binom{n}{k} \), defines the number of ways you can choose k items from a set of n items without considering the order of selection. It's a crucial part of calculating probabilities in a binomial distribution and can be calculated using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( n! \) represents the factorial of n. For instance, if we wish to find the number of ways to choose 5 snails out of 10, we calculate the binomial coefficient \( \binom{10}{5} \). This is essential in our probability calculation as it multiplies the likelihood of each combination occurring.

Probability Calculation

The probability calculation in a binomial distribution allows us to determine how likely an event is to occur over several trials. Using the earlier example of snails, we find out the chances of obtaining a certain number of streaked-shelled snails in a sample of ten. This is calculated using the binomial probability formula \( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \), where:\

\
• \( P(X=k) \) is the probability of getting exactly k successes (streaked shells) out of n trials (snails).\
• \( \binom{n}{k} \) is the binomial coefficient.\
• p is the probability of a single success on one trial.\
• \( (1-p) \) is the probability of a single failure.\
• k is the number of successes we're interested in.\
• n is the total number of trials.\

\
When we put our numbers into the equation, we conduct the probability calculation for the different scenarios provided by the exercise (a) 50%, (b) 60%, and (c) 70% streaked shells.

Random Sampling

Random sampling plays a crucial role in probability and statistics as it ensures that each member of a population has an equal chance of being selected, leading to an unbiased sample. In our scenario, choosing a random sample of 10 snails is critical to applying the binomial distribution method accurately. If the sample isn't random, the calculated probabilities won't represent the real-world situation correctly. The premise of random sampling allows us to use the binomial distribution model to anticipate the likelihood of various outcomes based on the proportion of snails with streaked shells in the broader population.