Question

60% of all students at a college still need to take another math class. If 49 students are randomly selected, find the probability that

a. Exactly 26 of them need to take another math class.____  
b. At most 31 of them need to take another math class.____   
c. At least 28 of them need to take another math class.____   
d. Between 25 and 31 (including 25 and 31) of them need to take another math class.____  

Answer 1

Mean = n * P = ( 49 * 0.6 ) = 29.4
Variance = n * P * Q = ( 49 * 0.6 * 0.4 ) = 11.76
Standard deviation = = 3.4293

Part a)

P ( X = 26 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 26 - 0.5 < X < 26 + 0.5 ) = P ( 25.5 < X < 26.5 )

P ( 25.5 < X < 26.5 )
Standardizing the value

Z = ( 25.5 - 29.4 ) / 3.4293
Z = -1.14
Z = ( 26.5 - 29.4 ) / 3.4293
Z = -0.85
P ( -1.14 < Z < -0.85 )
P ( 25.5 < X < 26.5 ) = P ( Z < -0.85 ) - P ( Z < -1.14 )
P ( 25.5 < X < 26.5 ) = 0.1989 - 0.1277
P ( 25.5 < X < 26.5 ) = 0.0712

Part b)

P ( X <= 31 )
Using continuity correction
P ( X < n + 0.5 ) = P ( X < 31 + 0.5 ) = P ( X < 31.5 )

P ( X < 31.5 )
Standardizing the value

Z = ( 31.5 - 29.4 ) / 3.4293
Z = 0.61

P ( X < 31.5 ) = P ( Z < 0.61 )
P ( X < 31.5 ) = 0.7291

Part c)

P ( X >= 28 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 28 - 0.5 ) =P ( X > 27.5 )

P ( X > 27.5 ) = 1 - P ( X < 27.5 )
Standardizing the value

Z = ( 27.5 - 29.4 ) / 3.4293
Z = -0.55

P ( Z > -0.55 )
P ( X > 27.5 ) = 1 - P ( Z < -0.55 )
P ( X > 27.5 ) = 1 - 0.2912
P ( X > 27.5 ) = 0.7088

Part d)

P ( 25 <= X <= 31 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 25 - 0.5 < X < 31 + 0.5 ) = P ( 24.5 < X < 31.5 )

P ( 24.5 < X < 31.5 )
Standardizing the value

Z = ( 24.5 - 29.4 ) / 3.4293
Z = -1.43
Z = ( 31.5 - 29.4 ) / 3.4293
Z = 0.61
P ( -1.43 < Z < 0.61 )
P ( 24.5 < X < 31.5 ) = P ( Z < 0.61 ) - P ( Z < -1.43 )
P ( 24.5 < X < 31.5 ) = 0.7299 - 0.0765
P ( 24.5 < X < 31.5 ) = 0.6533