The number of ways an unbiased coin lands tails either exactly 6 times or exactly 3 times when tossed 13 times is 59.9.
Binomial distribution gives a summary of the number of trials as well as observations as each trial has the same probability of attaining one particular value.
The unbiased coin is tossed 13 times.
p(6) = ₁₃C₆ × (0.5)⁶ × (0.5)¹³ ⁻ ⁶
p(3) = ₁₃C₃ × (0.5)³ × (0.5)¹³ ⁻ ³
P = p(9) + p(3)
P = ₁₃C₆ × (0.5)⁶ × (0.5)¹³ ⁻ ⁶ + ₁₃C₃ × (0.5)³ × (0.5)¹³ ⁻ ³
P = ₁₃C₆ × (0.5)¹³ + ₁₃C₃ × (0.5)¹³
P = (0.5)¹³ [₁₃C₆ + ₁₃C₃]
P = (0.5)¹³ × [ 13! / 6!( 7!) + 13! / 3!( 10! ) ]
P = (0.5)¹³ × [ 13! / 6!( 7!) + 13! / 3!( 10! ) ]
P = (0.5)¹³ [ 13 × 11 × 2 × 3 × 2 + 13 × 2 × 11 ]
P = 0.00012207031 × 2002
P = 0.24438476062
Hence, the unbiased coin can land tails either exactly 6 times or exactly 3 times in 59.9 ways.
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