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We need to determine the probability that the coin lands on tails at least 5 times in 6 tosses. What this means is that the coin will land on tails either 5 times out of 6 tosses or 6 times out of 6 tosses.

We can let T = the coin lands on tails on a single toss and H = the coin lands on heads on a single toss. Let’s now express the scenarios below and determine the probability of each one.

Scenario 1:

Out of 6 tosses, the coin lands on tails exactly 5 times.

We need to determine the number of possible outcomes of obtaining TTTTTH, not necessarily in that order.

TTTTTH can be arranged in $\frac{6!}{5!}=\text{6 ways}$, so there are six possible outcomes in which the coin lands on heads exactly five times.

Next, we can determine the probability of one possible outcome:

One possible outcome is TTTTTH, which has a probability of $\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}=\frac{1}{64}$.

Finally, to determine the actualprobability of the coin landing on heads exactly five times in six tosses, we calculate as follows:

$\Rightarrow$ number of possible outcomes producing the event × probability of just one outcome

$\Rightarrow 6\times \frac{1}{64}=\frac{6}{64}$

Now, let’s consider scenario 2.

Scenario 2:

Out of 6 tosses, the coin lands on tails exactly 6 times.

We need to determine the number of possible outcomes of TTTTTT. Notice that this outcome can occur in only one way. Thus, the probability is the following:

$\Rightarrow$ P(Scenario 1) + P(Scenario 2)

$\Rightarrow \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}=\frac{1}{64}$

Finally, to determine the probability that the coins land on heads at least five times in six tosses, we add the probabilities from scenario one and scenario two:

$\Rightarrow \frac{6}{64}+\frac{1}{64}=\frac{7}{64}$