In the above example, what would be the standard deviation of number of tails ?

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We start by identifying what the standard deviation of the number of tails in a binomial setting represents. If you flip a coin n times with tail probability p, the number of tails X ~ Binomial(n, p). The standard deviation is the square root of the variance, and Var(X) = n p (1 − p). Therefore, SD(X) = sqrt(n p (1 − p)).

Option 1: " + "n p (1 − p)" — This expression is precisely the variance, not the standard deviation. It omits the square root, so it underestimates by a factor of sqrt(n p (1 − p)) when you take the square root. This option is incorrect because it represents Var(X), not SD(X).

Option 2: " + "n p" — This is a common quantity in binomial contexts (the expected number of tails is n p), but it is not the standard deviation. It ignores the variability contributed by the (1 − p) term and so does not equal SD(X). This option is incorrect.

Option 3: " + "n p (1 − p)" — This is identical to Option 1, and as explained, it corresponds to the variance, not the standard deviation. The absence of the square root makes this choice incorrect for SD(X).

Option 4: " + "n (1 − p) 2" — Interpreted literally, this reads as n times (1 − p) squared, i.e., n (1 − p)^2. This does not equal n p (1 − p) inside a square root, nor does it reflect the correct variance or standard deviation formula. This option is incorrect because it mixes components in a way that does not correspond to Var(X) or SD(X).

Observation: The standard formula SD(X) = sqrt(n p (1 − p)) is not present among the given options. If allowed, the correct selection would reflect the sqrt of n p (1 − p). Since none of the options matches, the proper conclusion is that none of the listed choices correctly give the standard deviation; the missing correct form is sqrt(n p (1 − p)).

CVEN30008_2025_SM1 Module 6.2 Quiz

In the above example, what would be the standard deviation of number of tails ?

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