For one toss of a certain coin, the probability that the outcome is he

Veritas Prep Official Solution

This question is very similar to the questions we saw in the Probability book. The only difference is that we are not tossing a fair coin. The probability of getting heads is 0.7 not 0.5. So the probability of getting tails must be 0.3 since the total probability has to add up to 1.

The only acceptable cases are those in which we get ‘tails’ on all 6 tosses or we get tails on exactly 5 of the 6 tosses.

P(Tails on all 6 tosses) = (0.3)∗(0.3)∗(0.3)∗(0.3)∗(0.3)∗(0.3)=(0.3)6

P(Tails on exactly 5 tosses and Heads on one toss) = (0.3)^5∗(0.7)∗6

We multiply by 6 because 5 tails and 1 heads can be obtained in 6 different ways: HTTTTT, THTTTT, TTHTTT, TTTHTT, TTTTHT, TTTTTH

Probability that the outcome will be tails at least 5 times = Probability that the outcome will be tails 5 times + Probability that the outcome will be tails 6 times

Probability that the outcome will be tails at least 5 times = (0.3)^6+(0.3)^5∗(0.7)∗6

Again, the question is straight forward. It just has a little twist which sometimes throws people off during the test. It is these little things that differentiate a medium level question from a high level question.

P(H) = 7/10, therefore P(T) = 3/10

Out of 6 tosses we want at least 5 tails. = P(5 tails and 1 head) + P(6 tails)

P(5 tails and 1 head) = \(^6C_5 * (\frac{3}{10})^5 * \frac{7}{10} = 6 * \frac{243}{10^5} * \frac{7}{10} = \frac{10206}{10^6}\)


P(6 Tails) = \(^6C_6 * (\frac{3}{10})^6 = 1 * \frac{729}{10^6}\)


Therefore P(At least 5 tails) = \(\frac{10206}{10^6} + \frac{729}{10^6} = \frac{10935}{10^6} = 0.010935\)


Arun Kumar
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Solution:

If the outcome is at least 5 times tails in 6 tosses, then the outcome could be 5 times tails and 1 time heads OR all 6 times tails. Now let’s look at the probability of each option.

Option 1: 5 times tails and 1 time heads
P(TTTTTH) = 0.3 x 0.3 x 0.3 x 0.3 x 0.3 x 0.7 = 0.3^5 x 0.7

However, since the 5 T’s and 1 H can be arranged in 6!/5! = 6 ways, the probability is actually 6 x 0.3^5 x 0.7 = 4.2 x 0.3^5.

Option 2: 6 times tails

P(TTTTTT) = 0.3 x 0.3 x 0.3 x 0.3 x 0.3 x 0.3 = 0.3^6

Since the 6T’s can only be arranged in 6!/6! = 1 way, the probability is 0.3^6.

Therefore, the overall probability is 4.2 x 0.3^5 + 0.3^6 = 0.3^5 x (4.2 + 0.3) = 0.3^5 x 4.5.

Answer: 4.5 x 0.3^5

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