If a fair coin is tossed $7$ times, what is the probability, rounded to the nearest thousandth, of getting at most $1$ tails? $\newline$ Answer:

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Find probabilities using the binomial distribution

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Q. If a fair coin is tossed

7

times, what is the probability, rounded to the nearest thousandth, of getting at most

1

tails?

\newline

Answer:

Calculate Probability of $0$ Tails: We need to calculate the probability of getting at most $1$ tail in $7$ coin tosses. This includes the scenarios of getting exactly $0$ tails and exactly $1$ tail. Since the coin is fair, the probability of getting heads (H) or tails (T) on any single toss is $0.5$ .
Calculate Probability of $1$ Tail: First, let's calculate the probability of getting $0$ tails (which means getting $7$ heads). The probability of getting a head on one toss is $0.5$ , so for $7$ tosses it is $(0.5)^7$ .
Calculate Total Probability: Now, we calculate $(0.5)^7$ . $\newline$ $0.5^7 = 0.0078125$ $\newline$ This is the probability of getting $7$ heads ( $0$ tails) in $7$ tosses.
Round to Nearest Thousandth: Next, we calculate the probability of getting exactly $1$ tail. This can happen in $7$ different ways since the tail can appear in any one of the $7$ tosses. The probability for each of these ways is $(0.5)^6$ for the heads and $0.5$ for the one tail.
Round to Nearest Thousandth: Next, we calculate the probability of getting exactly $1$ tail. This can happen in $7$ different ways since the tail can appear in any one of the $7$ tosses. The probability for each of these ways is $(0.5)^6$ for the heads and $0.5$ for the one tail.Now, we calculate $(0.5)^6 \times 0.5$ . $\newline$ $0.5^6 \times 0.5 = 0.015625$ $\newline$ This is the probability of getting $1$ tail in a specific position among the $7$ tosses.
Round to Nearest Thousandth: Next, we calculate the probability of getting exactly $1$ tail. This can happen in $7$ different ways since the tail can appear in any one of the $7$ tosses. The probability for each of these ways is $(0.5)^6$ for the heads and $0.5$ for the one tail.Now, we calculate $(0.5)^6 \times 0.5$ . $\newline$ $0.5^6 \times 0.5 = 0.015625$ $\newline$ This is the probability of getting $1$ tail in a specific position among the $7$ tosses.Since there are $7$ different ways to get exactly $1$ tail, we multiply the probability of one specific way by $7$ . $\newline$ $7$ $2$ $\newline$ This is the probability of getting exactly $1$ tail in $7$ tosses.
Round to Nearest Thousandth: Next, we calculate the probability of getting exactly $1$ tail. This can happen in $7$ different ways since the tail can appear in any one of the $7$ tosses. The probability for each of these ways is $(0.5)^6$ for the heads and $0.5$ for the one tail.Now, we calculate $(0.5)^6 \times 0.5$ . $\newline$ $0.5^6 \times 0.5 = 0.015625$ $\newline$ This is the probability of getting $1$ tail in a specific position among the $7$ tosses.Since there are $7$ different ways to get exactly $1$ tail, we multiply the probability of one specific way by $7$ . $\newline$ $7 \times 0.015625 = 0.109375$ $\newline$ This is the probability of getting exactly $1$ tail in $7$ tosses.To find the total probability of getting at most $1$ tail, we add the probabilities of getting $0$ tails and $1$ tail. $\newline$ $7$ $1$ (probability of $0$ tails) $7$ $2$ $7$ $3$ (probability of $1$ tail) $7$ $4$
Round to Nearest Thousandth: Next, we calculate the probability of getting exactly $1$ tail. This can happen in $7$ different ways since the tail can appear in any one of the $7$ tosses. The probability for each of these ways is $(0.5)^6$ for the heads and $0.5$ for the one tail.Now, we calculate $(0.5)^6 \times 0.5$ . $0.5^6 \times 0.5 = 0.015625$ This is the probability of getting $1$ tail in a specific position among the $7$ tosses.Since there are $7$ different ways to get exactly $1$ tail, we multiply the probability of one specific way by $7$ . $7 \times 0.015625 = 0.109375$ This is the probability of getting exactly $1$ tail in $7$ tosses.To find the total probability of getting at most $1$ tail, we add the probabilities of getting $0$ tails and $1$ tail. $7$ $1$ (probability of $0$ tails) + $7$ $2$ (probability of $1$ tail) = $7$ $3$ Finally, we round the result to the nearest thousandth. $7$ $3$ rounded to the nearest thousandth is $7$ $5$ .