a fair is tossed 200 times. find the probability of getting a head an odd number of times.
Let Someone be the sample space of an experiment tossing a fair coin 200 times
ñ(S)=2^200
Let E be the event of getting an odd number of times
ñ(E)=200C1+200C3+.....+200C199=2^199
P(E)=ñ(E)/ñ(S) = 2^199/2^200 = 1/2
How the number of possible outcomes are 2^199
To find the probability of getting a head an odd number of times when a fair coin is tossed 200 times, we can use the concept of binomial probability.
The binomial probability formula is as follows:
P(x; n, P) = C(n, x) * P^x * (1 - P)^(n - x)
Where:
P(x; n, P) is the probability of obtaining exactly x successes in n trials.
C(n, x) is the number of combinations of n items taken x at a time.
P is the probability of success on a single trial.
In this case, we need to calculate the probability of getting a head (success) an odd number of times, so we can sum the probabilities of getting 1, 3, 5, 7, ..., up to 199 heads.
Let's break down the calculation into steps:
1. Calculate the number of trials, n = 200.
2. Calculate the probability of getting a head, P = 0.5 (since the coin is fair).
3. Calculate the probability for each odd number of heads, from 1 to 199.
P(heads = 1) = C(200, 1) * (0.5)^1 * (0.5)^(200-1)
P(heads = 3) = C(200, 3) * (0.5)^3 * (0.5)^(200-3)
...
P(heads = 199) = C(200, 199) * (0.5)^199 * (0.5)^(200-199)
4. Sum up the probabilities for all odd numbers of heads:
P(odd number of heads) = P(heads = 1) + P(heads = 3) + P(heads = 5) + ... + P(heads = 199)
You can use a calculator, spreadsheet software, or programming language to perform the calculations.
Keep in mind: Since the number of calculations required for this problem is large, using software can greatly simplify the process.
Pr(getting a head) =100/200
= 1/2
pr(of getting a head odd number) =17/200