Warner (1965) came up with an ingenious idea of estimating the population proportion of a sensitive characteristic by asking questions indirectly through a randomization device in face-to-face interviews. The point of this methodology is to ask sensitive questions in such a way that the respondents should not feel threatened while revealing truth about themselves. He uses a simple randomization device, such as a spinner, with two mutually exclusive outcomes; one related to the membership of a sensitive group and the other on its complement. If the outcome from the spinner matches with the status of the experiencing respondent, then he/she is asked to report “yes”. If the outcome from the spinner does not match with the status of the respondent, then he/she reports “no”. This simplicity within the respondent’s answer allows them to protect his/her privacy from the interviewer. Without a doubt, Warner’s (1965) idea has spread in the vast literature of survey sampling and has sprung the curiosity of many to be keen in developing an estimator better than his. Odumade and Singh (2009) came up with an idea of using a pair of deck of cards while collecting information on a sensitive characteristic in a face-to-face survey. The Odumade and Singh (2009) model has been observed to be an improvement over the Warner (1965) randomized response model and differs in the fact that it uses two randomization devices instead of one. In this thesis, we will further explore the possibilities of improving the Odumade and Singh (2009) model. A few new estimators of the population proportion will be developed and their properties such as unbiasedness, variances, and relative efficiency will be investigated analytically, empirically and through simulation studies. Lee, Sedory and Singh (2013) extended the idea to estimate the prevalence of two sensitive characteristics and their overlap. In this thesis we also consider the problem of estimating a proportion of two sensitive characteristics