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Class 10 Class 12

Probability

The probability of A hitting a target is

4 over 5

and that of B hitting it is

2 over 3.

They both fire at the target. Find the probability that
(i) at least one of them will hit the target.
(ii) only one of them will hit the target.

Let E denote the event ‘A hits the target’ and F denote the event ‘B hits the target’.

therefore

straight P left parenthesis straight E right parenthesis space equals 4 over 5 comma space space space straight P left parenthesis straight F right parenthesis space equals space 2 over 3

therefore space space space space space straight P left parenthesis straight E with bar on top right parenthesis space equals space 1 minus space straight P left parenthesis straight E right parenthesis space equals space 1 minus 4 over 5 space equals 1 fifth comma

straight P left parenthesis straight F with bar on top right parenthesis space equals space 1 minus straight P left parenthesis straight F right parenthesis space equals space 1 minus 2 over 3 space equals 1 third

(i) P(none of them hits the target) =

straight P left parenthesis straight E with bar on top space and space straight F with bar on top right parenthesis

equals space straight P open parentheses straight E with bar on top close parentheses space straight P open parentheses straight F with bar on top close parentheses space equals space 1 fifth cross times 1 third space equals 1 over 15

therefore space space space straight P left parenthesis at space least space one space of space them space hits space the space target right parenthesis space equals space 1 space minus space straight P left parenthesis none space of space them space hits space the space target right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 1 space minus space 1 over 15 space equals space 14 over 15

(ii) P(only one of them hits the target) =

straight P left parenthesis apostrophe straight E space and space straight F with bar on top apostrophe space space or space space straight E with bar on top space and space straight F apostrophe right parenthesis

equals space straight P open parentheses straight E space and space straight F with bar on top close parentheses space plus space straight P open parentheses straight E with bar on top space and space straight F close parentheses

equals space 4 over 5 cross times 1 third plus 2 over 3 cross times 2 over 5 space equals space 4 over 15 plus 4 over 15 space equals space 8 over 15

73 Views

The probability that A will solve a problem is

2 over 5

and B will solve it is

5 over 6

. If both try the problem independently, find the probability that the problem will be solved.

The probabilities of A and B solving the problem are $2 over 5 space and space 5 over 6$ respectively.
$therefore$ the probabilities of A and B not solving the problems are $1 minus 2 over 5 comma space 1 space minus 5 over 6$ i.e., $3 over 5 comma space space 1 over 6$ respectively.

$therefore$ probabilities that problem is not solved by A and B $equals space 3 over 5 cross times 1 over 6 space equals 1 over 10$
$therefore$ probability that the problem will be solved = $1 minus 1 over 10 space equals space 9 over 10$

90 Views

A problem of Mathematics is given to three students, whose chances of solving it are

1 third comma space 1 fourth comma space 1 fifth.

What is the probability that the problem will be solved?

The probabilities of three students solving the problem are $1 third comma space 1 fourth comma space 1 fifth$ respectively.
$therefore$ proabilites of three students not solving the problem are
$1 minus 1 third space equals 2 over 3 comma space space 1 minus 1 fourth space equals 3 over 4 comma space space 1 minus 1 fifth space equals 4 over 5 space respectively.$
$therefore$ the probability that the problem is not solved by any one of them
$equals space 2 over 3 cross times 3 over 4 cross times 4 over 5 space equals 2 over 5$
$therefore$ probability that the problem will be solved by atleast one of them
$equals space 1 minus 2 over 5 space equals space 3 over 5$

82 Views

A certain team wins with probability 0.7, losses with probability 0.2 and ties with probability 0.1. The team plays three games. Find the probability that the team wins at least two of the games, but not lose.

Let W , L and T denote the events that the team wins, loses and ties respectively.
∴ P (W) = 0.7, P (L) = 0.2, P (T) = 0.1
Required probability = P (team wins at least two games but not loses)
= P (WWT) + P (WTW) + P (TWW) + P (WWW)
= P (W) P (W) P (T) + P (W) P (T) P (W) + P (T) P (W) P (W) + P(W) P(W) P(W)
= 3 [P (W) P (W) P (T)] + P (W) P (W) P (W)
= 3 [(0.7) (0.7) (0.1) + (0.7) (0.7) (0.7)
= 3 × 0.049 + 0.343 = 0.147 + 0.343
= 0.49

254 Views