Lucas Delgado
The probability that out of 175 chicks hatched on Peeper Farm, a certain number will be female is a common probability question. The question usually involves assumptions that male and female chicks are equally probable and asks for the probability of a certain number, or a number above a certain threshold, being female. This can be calculated using the binomial distribution, converting the problem into a standard normal distribution (z-score) and looking up the z-score in a standard normal distribution table.
| Characteristics | Values |
|---|---|
| Total number of chicks | 175 |
| Number of female chicks | 80 or 90 |
| Probability of a chick being female | 0.45 or 45% |
| Probability of at least 80 female chicks | 88.7% |
| Probability of at least 90 female chicks | 38.2% or 99.7% |
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What You'll Learn
Calculating probability
To calculate the probability of an event occurring, we can use the formula: Probability = number of favourable outcomes / total number of outcomes. This can also be defined as the ratio of favourable outcomes to the total number of events.
For example, consider the question: "What is the probability that out of 175 chicks hatched on Peeper Farm, at least 80 will be female, assuming males and females are equally probable?". Here, the favourable outcome is the number of female chicks, and the total number of outcomes is 175 (the total number of chicks).
Using the formula, we get: Probability = number of female chicks / total number of chicks = 80 / 175 = 0.45 or 45%. This is the probability of having exactly 80 female chicks.
However, the question asks for the probability of having "at least 80" female chicks. To calculate this, we can use the binomial probability formula, as each chick has an equal chance of being male or female. The parameters for the binomial distribution are:
- Number of trials (n): 175 (total number of chicks)
- Probability of success (p): 0.5 (probability of a chick being female)
- Number of successes (k): 80 or more (we want at least 80 females)
Using the normal approximation of the binomial distribution, we can calculate the probability of having at least 80 female chicks as approximately 88.7%.
Similarly, we can calculate the probability of other scenarios, such as having at least 90 female chicks out of 175. Using the binomial distribution and the normal approximation, the probability is found to be approximately 38.2%.
In summary, to calculate probabilities, we use specific formulas depending on the nature of the problem. For basic probability calculations, we use the ratio of favourable outcomes to total outcomes. For more complex scenarios, such as those involving a large number of trials, we may use the binomial distribution and its normal approximation.
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Binomial distribution
In the context of the given problem, we are interested in finding the probability that out of 175 chicks, at least 80 or 90 will be female. This can be solved using binomial distribution. The parameters for the binomial distribution in this case are:
- Number of trials (n): 175 (the total number of chicks)
- Probability of success (p): 0.5 (assuming the probability of a chick being female is equal to the probability of it being male)
- Number of successes (k): 80 or 90 (depending on the problem variation)
Using these parameters, we can apply the binomial probability formula to calculate the probability of getting exactly k successes in n independent trials. The formula is given by:
> {\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}}
This formula can be understood as the probability of obtaining k successes and n-k failures in a sequence of n independent Bernoulli trials, where each trial has a probability p of success.
For the problem with at least 80 female chicks, the normal approximation of the binomial distribution yields a probability of approximately 88.7%. For the problem with at least 90 female chicks, the binomial cumulative distribution function gives a value of 99.7%.
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Standard normal distribution
The standard normal distribution, also known as the z-distribution, is a probability distribution that is often used to approximate binomial distributions when the number of trials is large. It is a symmetric distribution with a mean of 0 and a standard deviation of 1.
In the context of the problem "What is the probability that out of 175 chicks, at least 80 will be female?", the standard normal distribution can be applied. First, we calculate the mean (μ) and standard deviation (σ) of the binomial distribution, assuming a probability of 0.5 for a chick being female. The mean is 87.5 and the standard deviation is approximately 6.6.
Next, we convert the problem into a standard normal distribution (z-score) by subtracting the mean from the number of females and dividing by the standard deviation. In this case, the z-score for 80 females is calculated as (80-87.5)/6.6, which is approximately -1.14. Due to the continuity correction, we use the z-score for 79.5 females, which is -1.21.
Finally, we look up the z-score in a standard normal distribution table. The probability of z being less than -1.21 is approximately 0.1131. Thus, the probability of having at least 80 females is 1 - 0.1131, which is approximately 88.7%.
The standard normal distribution is useful in many real-world applications, such as calculating the percentage of people with an IQ between two values or the likelihood of certain heights in a population. It provides a standardised framework to estimate probabilities and make predictions based on normally distributed data.
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Z-score
The problem states that there are 175 chicks and asks for the probability that at least 80 or 90 of them will be female. This can be solved using the concept of z-scores.
In the context of the chick problem, the z-score can be used to determine the probability of having at least 80 or 90 female chicks. The first step is to calculate the mean and standard deviation. The mean (μ) of a binomial distribution is calculated as μ=n⋅p, where n is the total number of chicks (175) and p is the probability of success (0.5) since the probability of a chick being female or male is assumed to be equal. The standard deviation (σ) is calculated as σ=n⋅p⋅(1−p).
For the problem with at least 80 female chicks, the z-score for k = 79.5 (continuity correction applied) is calculated as z=σk−μ=6.679.5−87.5≈−1.21. Using the standard normal distribution table, the probability of z being less than -1.21 is found to be approximately 0.1131. Thus, the probability of having at least 80 females is approximately 88.7%.
For the problem with at least 90 female chicks, the z-score is calculated with the continuity correction as P(X≥90)≈P(X≥89.5). The z-score is z=σx−μ=6.689.5−87.5≈0.303. Using the standard normal distribution table, the probability for a z-score of approximately 0.303 is about 0.618. Therefore, the probability of having at least 90 female chicks is 1−0.618 = 0.382 or 38.2%.
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Binomial probability formula
The binomial probability formula is used to calculate the probability of achieving a specific number of successes in a fixed number of independent trials or experiments, each of which has only two possible outcomes. The formula is:
P(X = x) = nCx * px * (1 - p)n-x
Where:
- P(X = x) is the probability of achieving exactly x successes.
- N is the total number of trials or experiments.
- X is the number of successes being sought.
- P is the probability of success in a single trial.
- NCx is the binomial coefficient, also known as the combination "n choose x," and represents the number of ways to choose x items from a set of n items without regard to the order.
For example, consider the question, "What is the probability that out of 175 chicks hatched, at least 80 will be female?" Assuming that the probability of a chick being male or female is equal, we can use the binomial probability formula to calculate the desired probability.
First, we define the parameters:
- Number of trials (n) = 175 (total number of chicks)
- Probability of success (p) = 0.5 (probability of a chick being female)
- Number of successes (x) = 80 or more (seeking at least 80 females)
Next, we can calculate the probability using the formula:
P(X ≥ 80) = nCx * px * (1 - p)n-x
Using this formula, we can determine the probability of having at least 80 female chicks out of 175. However, due to the large value of n, it is more practical to use the normal approximation of the binomial distribution or a statistical table/calculator with a binomial cumulative distribution function (BCDF). Applying these methods yields a probability of approximately 88.7% for having at least 80 female chicks.
In another example, we can consider the question, "What is the probability that out of 175 chicks hatched, at least 90 will be female?" Using the same assumptions and methods as before, we find that the probability of having at least 90 female chicks out of 175 is approximately 38.2%.
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Frequently asked questions
88.7%.
38.2%.
99.7%.
