A coin was tossed 1000 times in an experiment to decide whether it really had an equal chance of coming up heads or tails. The decision was made in advance to declare the coind NOT FAIR if the observed outcome of the experiment had less than a 5% chance of happening. Would the coin be declared not fair if fewer than 470 of the 1000 tosses came up heads?
To determine whether the coin would be declared not fair if fewer than 470 of the 1000 tosses came up heads, we need to calculate the probability of getting fewer than 470 heads assuming that the coin is fair.
Assuming the coin is fair, the probability of getting heads on a single toss is 0.5. We can use the binomial distribution formula to calculate the probability of getting fewer than 470 heads when tossing the coin 1000 times.
The formula to calculate the probability of getting exactly k successes (or heads in this case) in n trials (tosses) is:
P(X=k) = (nCk) * p^k * (1-p)^(n-k)
where:
- n is the number of trials (1000 tosses)
- k is the number of successes (number of heads)
- p is the probability of success on a single trial (0.5)
Now we need to calculate the probability of getting fewer than 470 heads, which means we need to sum up the probabilities of getting 0, 1, 2, ..., 469 heads.
P(X < 470) = P(X=0) + P(X=1) + P(X=2) + ... + P(X=469)
Let's calculate this probability step by step:
Step 1: Calculate the probability of getting exactly k heads in a single trial:
P(X=k) = (1000Ck) * (0.5)^k * (1-0.5)^(1000-k)
Step 2: Sum up the probabilities of getting 0 to 469 heads:
P(X < 470) = P(X=0) + P(X=1) + P(X=2) + ... + P(X=469)
Let's calculate P(X < 470) using a Python script:
```python
from math import comb
n = 1000
p = 0.5
prob = 0
for k in range(470):
prob += comb(n, k) * p**k * (1-p)**(n-k)
prob
```
After running the code, we find that the probability of getting fewer than 470 heads in 1000 tosses is approximately 0.028.
Since this probability (0.028) is less than the threshold of 0.05 (5%), the observed outcome of getting fewer than 470 heads would be considered statistically significant and provide evidence to declare the coin as NOT FAIR.
To determine whether the coin would be declared not fair, we need to calculate the probability of observing fewer than 470 heads out of 1000 tosses.
The probability of getting heads on a fair coin toss is 0.5 (or 50%). If the coin truly has an equal chance of coming up heads or tails, we would expect to see approximately 500 heads out of 1000 tosses.
To calculate the probability of observing fewer than 470 heads, we can use the binomial probability formula. Let's assume X is the random variable representing the number of heads obtained in 1000 tosses. The binomial distribution is defined as B(n, p), where n is the number of trials (1000 tosses) and p is the probability of success (0.5).
P(X < 470) = P(X = 0) + P(X = 1) + ... + P(X = 469)
We can use a statistical tool, such as a binomial probability calculator or a software package (like R or Excel), to calculate this probability. Alternatively, we can use a normal approximation to the binomial distribution when the sample size (1000 tosses) is large enough.
Using the normal approximation approach, first, we calculate the mean (µ) and standard deviation (σ) of the binomial distribution:
µ = n * p = 1000 * 0.5 = 500
σ = sqrt(n * p * (1 - p)) = sqrt(1000 * 0.5 * 0.5) = sqrt(250) ≈ 15.81
Next, we standardize the value of 470 using the z-score formula:
z = (x - µ) / σ = (470 - 500) / 15.81 ≈ -1.893
By looking up the z-score in a standard normal distribution table or using a calculator, we can find the corresponding probability:
P(X < 470) ≈ P(Z < -1.893) ≈ 0.029
Comparing this probability to the pre-defined threshold of 5% (0.05), we see that the observed outcome has a probability of less than 5%. Therefore, if fewer than 470 heads came up, the coin would be declared not fair based on the predetermined decision criteria.