Three coins are tossed simultaneously $100$ times with the following frequencies of different outcomes:
OUTCOME No Head One Head Two Heads Three Heads
FREQUENCY $14$ $38$ $36$ $12$
If the three coins are simultaneously tossed again, compute the probability of (i) Two heads (ii) Three heads (iii) at least one head (iv) more heads than tails (v) more tails than heads.

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Solution

Given,
Total number of trials $= 100$
Number of No Head $= 14$
Number of One head $= 38$
Number of Two heads $= 36$
Number of Three heads $= 12$
We use the formula,
Probability of favorable outcome $= \frac{N u m b e r o f f a v o r a b l e o u t c o m e s}{T o t a l n u m b e r o f t r i a l s}$
STEP 1: Find the probability of two heads
Probability of Two heads $= \frac{N u m b e r o f t w o h e a d s}{T o t a l n u m b e r o f t r i a l s} = \frac{36}{100} = 0.36$
STEP 2: Find the probability of three heads
Probability of Three heads $= \frac{N u m b e r o f t h r e e h e a d s}{T o t a l n u m b e r o f t r i a l s} = \frac{12}{100} = 0.12$
STEP 3: Find the probability of at least one head
Probability of at least one head $= \frac{N u m b e r o f o n e h e a d + N u m b e r o f t w o h e a d s + N u m b e r o f t h r e e h e a d s}{T o t a l n u m b e r o f t r i a l s} = \frac{38 + 36 + 12}{100} = \frac{86}{100} = 0.86$
STEP 4: Find the probability of more heads than tails
Probability of more heads than tails $= \frac{N u m b e r o f (3 h e a d s, 0 t a i l) + N u m b e r o f (2 h e a d s, 1 t a i l)}{T o t a l n u m b e r o f t r i a l s}$
$= \frac{12 + 36}{100} = \frac{48}{100} = 0.48$
STEP 5: Find the probability of more tails than heads
Probability of more tails than heads $= \frac{N u m b e r o f (3 t a i l s, 0 h e a d) + N u m b e r o f (2 t a i l s, 1 h e a d)}{T o t a l n u m b e r o f t r i a l s}$
$= \frac{14 + 38}{100} = \frac{52}{100} = 0.52$
Hence, the probability of (i) Two heads is $0.36$ (ii) Three heads is $0.12$ (iii) at least one head is $0.86$ (iv) more heads than tails is $0.48$ (v) more tails than heads is $0.52$

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OUTCOME	No Head	One Head	Two Heads	Three Heads
FREQUENCY	$14$	$38$	$36$	$12$