A coin is tossed 12 times. What is the probability of getting exactly 6 tails?

Question

What is the probability of getting 6 consecutive tails?

David Armstrong · Accepted Answer

The probability is #P=924/4096~~0.2256#.

Let #P# be the probability of getting exactly #6# tails, when a coin is tossed #12# times. Now let us consider the following: probability of success from one toss be #color(red)(p)#. probability of failure from one toss be #color(red)(q)#. number of trials be #color(red)(n)#. number of success be #color(red)(r)#. number of failures is #color(red)(n-r)#. Hence, the total probability of succeeding is represented by #rarr# #color(red)(P=nC_(r)p^(r)q^(r-x))#.......(1). Here, #p=q=1/2# #n=12# #larr# Given. #r=6# #larr# Given. #:.n-r=6# Now, substituting this into (1), we get #P=""_12C_6(1/2)^6(1/2)^6# #color(white)P=(12!)/(6!(12-6)!)*1/2^12# #color(white)P=924*1/4096# #color(white)P~~0.2256# Hope it Helps :)

Quinn Allen · Answer

undefined To find the probability of getting exactly 6 tails when a coin is tossed 12 times, you can use the binomial probability formula, which is:

$$ P(X = k) = \binom{n}{k} \times p^k \times (1-p)^{n-k} $$

Where:
- $ n $ is the number of trials (in this case, 12 tosses).
- $ k $ is the number of successes (in this case, 6 tails).
- $ p $ is the probability of success on each trial (for a fair coin, $ p = 0.5 $).
- $ \binom{n}{k} $ represents the number of combinations of $ n $ items taken $ k $ at a time.

Substitute the values into the formula:

$$ P(X = 6) = \binom{12}{6} \times (0.5)^6 \times (1-0.5)^{12-6} $$

$$ P(X = 6) = \binom{12}{6} \times (0.5)^6 \times (0.5)^6 $$

Calculate:

$$ P(X = 6) = \binom{12}{6} \times (0.5)^{12} $$

$$ P(X = 6) = \frac{12!}{6!(12-6)!} \times 0.5^{12} $$

$$ P(X = 6) = \frac{479001600}{720 \times 720} \times 0.5^{12} $$

$$ P(X = 6) = \frac{479001600}{518400} \times 0.5^{12} $$

$$ P(X = 6) ≈ 0.2256 $$

So, the probability of getting exactly 6 tails when a coin is tossed 12 times is approximately 0.2256.

Emery Adams · Answer

undefined To find the probability of getting exactly 6 tails when a coin is tossed 12 times, you can use the binomial probability formula.

The binomial probability formula is:

$$ P(X = k) = \binom{n}{k} \times p^k \times (1-p)^{n-k} $$

Where:
- $ n $ is the total number of trials (in this case, the number of coin tosses)
- $ k $ is the number of successes (in this case, the number of tails)
- $ p $ is the probability of success on each trial (for a fair coin, $ p = 0.5 $)
- $ \binom{n}{k} $ is the number of combinations of $ n $ items taken $ k $ at a time (this represents the number of ways to get exactly $ k $ tails in $ n $ tosses)

Substituting the given values into the formula:

$$ P(X = 6) = \binom{12}{6} \times (0.5)^6 \times (1-0.5)^{12-6} $$

$$ P(X = 6) = \binom{12}{6} \times (0.5)^6 \times (0.5)^6 $$

$$ P(X = 6) = \binom{12}{6} \times (0.5)^{12} $$

Now, calculate $ \binom{12}{6} $:

$$ \binom{12}{6} = \frac{12!}{6!(12-6)!} = \frac{12!}{6!6!} $$

$$ = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6 \times 5 \times 4 \times 3 \times 2 \times 1} $$

$$ = 924 $$

Now, substitute this value into the equation:

$$ P(X = 6) = 924 \times (0.5)^{12} $$

$$ P(X = 6) \approx 0.2256 $$

So, the probability of getting exactly 6 tails when a coin is tossed 12 times is approximately 0.2256, or about 22.56%.