If we flip a coin three times what is the theoretical probability of getting zero heads, one heads, two heads, and three heads PLEASE ANSWER
P(0)=P(3) = (1/2)^3
P(1)=P(2) = 3*(1/2)^3
Because 3C1 = 3C2 = 3
p heads = p tails = 0.5
lets call p heads favorable = p = .5
p tails unfavorable = q = 1-p = .5
P r times in n trials = C(n,r) p^r q^(n-r) where C(n,r) = n!/[r!(n-r)! defined = 1 if r = 0
so here n = 3
Zero heads
n = 3, r = 0
P = 1 * .5^0 * .5^(3) = .5^3 = .125 or 1/8
by the way that is the same as zero taails = 3 heads :)
now the hard one
Two heads, one tail
n = 3, r = 2
C(3,2) = 3!/ [ 2!(1!)] = 3
P = 3 * .5^2 * .5^1 = 3* .125 = 3/8
To find the theoretical probability of getting zero, one, two, or three heads when flipping a coin three times, we need to consider the total number of possible outcomes and the number of favorable outcomes for each case.
1. Probability of getting zero heads:
When flipping a coin, there are two possible outcomes - heads or tails. So, for each coin flip, there are two possibilities. Since we are flipping the coin three times, the total number of possible outcomes is 2 x 2 x 2 = 8.
The favorable outcome for zero heads is only one possibility - getting tails on all three flips.
Therefore, the probability of getting zero heads is 1/8.
2. Probability of getting one head:
Similar to the previous case, there are two possible outcomes for each coin flip. The order of the events matters here, so we need to consider the different ways of arranging the one head within the three flips.
The favorable outcomes for one head are: H-T-T, T-H-T, or T-T-H.
The probability of getting one head is 3/8.
3. Probability of getting two heads:
As above, there are two possible outcomes for each coin flip, and the order is important. We need to consider the different arrangements of the two heads within the three flips.
The favorable outcomes for two heads are: H-H-T, H-T-H, or T-H-H.
The probability of getting two heads is 3/8.
4. Probability of getting three heads:
Again, there are two possible outcomes for each coin flip, and the order matters.
The favorable outcome for three heads is only one possibility - getting heads on all three flips.
Therefore, the probability of getting three heads is 1/8.
To summarize:
- Probability of getting zero heads: 1/8
- Probability of getting one head: 3/8
- Probability of getting two heads: 3/8
- Probability of getting three heads: 1/8
To calculate the theoretical probability of flipping a coin three times and getting different outcomes, we need to determine the total number of possible outcomes and the number of favorable outcomes for each outcome.
The total number of possible outcomes when flipping a coin three times can be calculated using the multiplication rule. Since each coin flip has two possible outcomes (heads or tails), the total number of possible outcomes is 2 x 2 x 2 = 8.
Now let's determine the number of favorable outcomes for each outcome:
1. Zero heads: This means getting three tails. There is only one favorable outcome for this, which is TTT. So, the number of favorable outcomes for zero heads is 1.
2. One head: This means getting one head and two tails. There are three possible positions for the head (HTT, THT, or TTH). So, the number of favorable outcomes for one head is 3.
3. Two heads: This means getting two heads and one tail. Again, there are three possible positions for the tails (HHT, HTH, or THH). So, the number of favorable outcomes for two heads is also 3.
4. Three heads: This means getting three heads and no tails. There is only one favorable outcome for this, which is HHH. So, the number of favorable outcomes for three heads is 1.
Now, let's calculate the theoretical probability for each outcome by dividing the number of favorable outcomes by the total number of possible outcomes:
1. Probability of zero heads: 1 / 8 = 0.125 or 12.5%
2. Probability of one head: 3 / 8 = 0.375 or 37.5%
3. Probability of two heads: 3 / 8 = 0.375 or 37.5%
4. Probability of three heads: 1 / 8 = 0.125 or 12.5%
So, the theoretical probability of getting zero heads, one head, two heads, and three heads when flipping a coin three times are 0.125, 0.375, 0.375, and 0.125, respectively.