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 if 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.
To do this, we can use the binomial distribution formula:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
where:
- P(X = k) is the probability of getting exactly k successes (in this case, heads)
- 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 for a fair coin)
- C(n, k) is the binomial coefficient, which represents the number of ways to choose k successes from n trials
In this case, we want to calculate the probability of getting fewer than 470 heads, so we need to sum up the probabilities of getting 0, 1, 2, ..., 469 heads.
The probability of getting exactly k heads can be calculated using the formula above. We can then sum up the probabilities of getting 0 to 469 heads to determine the overall probability.
Let's use a calculator or statistical software to perform this calculation.
Using a binomial calculator, the probability of getting fewer than 470 heads with a fair coin can be calculated as follows:
P(X < 470) = Σ [P(X = k)] for k = 0 to 469
The result is approximately 0.025, or 2.5%.
Since the calculated probability is less than 5%, the observed outcome of getting fewer than 470 heads has less than a 5% chance of happening assuming a fair coin.
Therefore, if fewer than 470 of the 1000 tosses came up heads, the coin would be declared not fair according to the pre-set criteria.
To determine whether the coin would be declared not fair if fewer than 470 tosses came up heads out of 1000 tosses, we need to analyze the probability of this occurring.
The first step is to calculate the probability of getting heads in a single coin toss. Assuming the coin is fair, the probability of getting heads is 0.5 (50%), since there are only two possible outcomes - heads or tails - and they have an equal chance of occurring.
Next, we use the binomial distribution formula to find the probability of a specific number of heads occurring in multiple tosses. In this case, we want to find the probability of getting fewer than 470 heads out of 1000 tosses.
Using a statistical tool or a calculator that provides binomial distribution calculations, we can calculate the probability of observing fewer than 470 heads. If this probability is less than 5%, then the coin would be declared not fair.
Keep in mind that the exact calculation might take some time, so using a calculator or statistical software is recommended. But, to give you an idea, here's how the calculation could be approached in Python:
```python
import scipy.stats as stats
# Probability of getting heads in a single toss
p = 0.5
# Total number of tosses
n = 1000
# Probability of getting fewer than 470 heads
prob = sum(stats.binom.pmf(k, n, p) for k in range(470))
# Checking if the probability is less than 5%
if prob < 0.05:
print("The coin would be declared not fair.")
else:
print("The coin would not be declared not fair.")
```
By calculating the probability, we can determine whether the coin would be declared not fair if fewer than 470 heads were observed in the experiment.