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Probability — Applied Maths STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceApplied MathsProbability5 Marks
Question
What is the probability that a standard normal variate Z will be
(i) greater than 1.09
(ii) less than - 1.65
(iii) lying between - 1.00 and 1.96
✓

Answer

(i) greater than 1.09
The total area under the curve is equal to 1, so that the total area to the right Z = 0 is 0.5 (since the curve is symmetrical). The area between Z = 0 and 1.09 (from tables) is 0.3621.
Image
$P(Z > 1.09) = 0.5000 - 0.3621$
$= 0.1379$
The shaded area to the right of Z = 1.09 is the probability that Z will be greater than 1.09.
(ii) less than -1.65
The area between -1.65 and 0 is the same as area between 0 and 1.65. In the table the area between zero and 1.65 is 0.4505 (from the table). Since the area to the left of zero is $0.5, P(Z < 1.65) =0.5000-0.4505 =0.0495.$
Image
(iii) lying between -1.00 and 1.96
The probability that the random variable Z in between -1.00 and 1.96 is found by adding the corresponding areas:
Image
Area between -1.00 and 1.96 = area between (-1.00 and 0) + area between (0 and 1.96)
$P(- 1.00 < Z < 1.96) = P(- 1.00 < Z < 0) + P(0 < Z < 1.96)$
$= 0.3413+0.4750$ (by tables)
$= 0.8163$

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